Changed derivative of abs(x) to
$$
\frac{\overline{x}}{2\ abs(x)}
$$
Added conjugate(x).
\begin{spad}
)abbrev package FSPECX FunctionalSpecialFunction
)boot $tryRecompileArguments := nil
++ Provides the special functions
++ Author: Manuel Bronstein
++ Date Created: 18 Apr 1989
++ Date Last Updated: 4 October 1993
++ Description: Provides some special functions over an integral domain.
++ Keywords: special, function.
FunctionalSpecialFunction(R, F) : Exports == Implementation where
R : Join(Comparable, IntegralDomain)
F : FunctionSpace R
OP ==> BasicOperator
K ==> Kernel F
SE ==> Symbol
SPECIALDIFF ==> '%specialDiff
Exports ==> with
belong? : OP -> Boolean
++ belong?(op) is true if op is a special function operator;
operator : OP -> OP
++ operator(op) returns a copy of op with the domain-dependent
++ properties appropriate for F;
++ error if op is not a special function operator
abs : F -> F
++ abs(f) returns the absolute value operator applied to f
conjugate: F -> F
++ conjugate(f) returns the conjugate value operator applied to f
Gamma : F -> F
++ Gamma(f) returns the formal Gamma function applied to f
Gamma : (F, F) -> F
++ Gamma(a, x) returns the incomplete Gamma function applied to a and x
Beta : (F, F) -> F
++ Beta(x, y) returns the beta function applied to x and y
digamma : F->F
++ digamma(x) returns the digamma function applied to x
polygamma : (F, F) ->F
++ polygamma(x, y) returns the polygamma function applied to x and y
besselJ : (F, F) -> F
++ besselJ(x, y) returns the besselj function applied to x and y
besselY : (F, F) -> F
++ besselY(x, y) returns the bessely function applied to x and y
besselI : (F, F) -> F
++ besselI(x, y) returns the besseli function applied to x and y
besselK : (F, F) -> F
++ besselK(x, y) returns the besselk function applied to x and y
airyAi : F -> F
++ airyAi(x) returns the Airy Ai function applied to x
airyAiPrime : F -> F
++ airyAiPrime(x) returns the derivative of Airy Ai function applied to x
airyBi : F -> F
++ airyBi(x) returns the Airy Bi function applied to x
airyBiPrime : F -> F
++ airyBiPrime(x) returns the derivative of Airy Bi function applied to x
lambertW : F -> F
++ lambertW(x) is the Lambert W function at x
polylog : (F, F) -> F
++ polylog(s, x) is the polylogarithm of order s at x
weierstrassP : (F, F, F) -> F
++ weierstrassP(g2, g3, x)
weierstrassPPrime : (F, F, F) -> F
++ weierstrassPPrime(g2, g3, x)
weierstrassSigma : (F, F, F) -> F
++ weierstrassSigma(g2, g3, x)
weierstrassZeta : (F, F, F) -> F
++ weierstrassZeta(g2, g3, x)
-- weierstrassPInverse : (F, F, F) -> F
-- ++ weierstrassPInverse(g2, g3, z) is the inverse of Weierstrass
-- ++ P function, defined by the formula
-- ++ \spad{weierstrassP(g2, g3, weierstrassPInverse(g2, g3, z)) = z}
whittakerM : (F, F, F) -> F
++ whittakerM(k, m, z) is the Whittaker M function
whittakerW : (F, F, F) -> F
++ whittakerW(k, m, z) is the Whittaker W function
angerJ : (F, F) -> F
++ angerJ(v, z) is the Anger J function
weberE : (F, F) -> F
++ weberE(v, z) is the Weber E function
struveH : (F, F) -> F
++ struveH(v, z) is the Struve H function
struveL : (F, F) -> F
++ struveL(v, z) is the Struve L function defined by the formula
++ \spad{struveL(v, z) = -%i^exp(-v%pi%i/2)struveH(v, %iz)}
hankelH1 : (F, F) -> F
++ hankelH1(v, z) is first Hankel function (Bessel function of
++ the third kind)
hankelH2 : (F, F) -> F
++ hankelH2(v, z) is the second Hankel function (Bessel function of
++ the third kind)
lommelS1 : (F, F, F) -> F
++ lommelS1(mu, nu, z) is the Lommel s function
lommelS2 : (F, F, F) -> F
++ lommelS2(mu, nu, z) is the Lommel S function
kummerM : (F, F, F) -> F
++ kummerM(a, b, z) is the Kummer M function
kummerU : (F, F, F) -> F
++ kummerU(a, b, z) is the Kummer U function
legendreP : (F, F, F) -> F
++ legendreP(nu, mu, z) is the Legendre P function
legendreQ : (F, F, F) -> F
++ legendreQ(nu, mu, z) is the Legendre Q function
kelvinBei : (F, F) -> F
++ kelvinBei(v, z) is the Kelvin bei function defined by equality
++ \spad{kelvinBei(v, z) = imag(besselJ(v, exp(3%pi%i/4)z))}
++ for z and v real
kelvinBer : (F, F) -> F
++ kelvinBer(v, z) is the Kelvin ber function defined by equality
++ \spad{kelvinBer(v, z) = real(besselJ(v, exp(3%pi%i/4)z))}
++ for z and v real
kelvinKei : (F, F) -> F
++ kelvinKei(v, z) is the Kelvin kei function defined by equality
++ \spad{kelvinKei(v, z) =
++ imag(exp(-v%pi%i/2)besselK(v, exp(%pi%i/4)z))}
++ for z and v real
kelvinKer : (F, F) -> F
++ kelvinKer(v, z) is the Kelvin kei function defined by equality
++ \spad{kelvinKer(v, z) =
++ real(exp(-v%pi%i/2)besselK(v, exp(%pi%i/4)z))}
++ for z and v real
ellipticK : F -> F
++ ellipticK(m) is the complete elliptic integral of the
++ first kind: \spad{ellipticK(m) =
++ integrate(1/sqrt((1-t^2)(1-mt^2)), t = 0..1)}
ellipticE : F -> F
++ ellipticE(m) is the complete elliptic integral of the
++ second kind: \spad{ellipticE(m) =
++ integrate(sqrt(1-mt^2)/sqrt(1-t^2), t = 0..1)}
ellipticE : (F, F) -> F
++ ellipticE(z, m) is the incomplete elliptic integral of the
++ second kind: \spad{ellipticE(z, m) =
++ integrate(sqrt(1-mt^2)/sqrt(1-t^2), t = 0..z)}
ellipticF : (F, F) -> F
++ ellipticF(z, m) is the incomplete elliptic integral of the
++ first kind : \spad{ellipticF(z, m) =
++ integrate(1/sqrt((1-t^2)(1-mt^2)), t = 0..z)}
ellipticPi : (F, F, F) -> F
++ ellipticPi(z, n, m) is the incomplete elliptic integral of
++ the third kind: \spad{ellipticPi(z, n, m) =
++ integrate(1/((1-nt^2)sqrt((1-t^2)(1-mt^2))), t = 0..z)}
jacobiSn : (F, F) -> F
++ jacobiSn(z, m) is the Jacobi elliptic sn function, defined
++ by the formula \spad{jacobiSn(ellipticF(z, m), m) = z}
jacobiCn : (F, F) -> F
++ jacobiCn(z, m) is the Jacobi elliptic cn function, defined
++ by \spad{jacobiCn(z, m)^2 + jacobiSn(z, m)^2 = 1} and
++ \spad{jacobiCn(0, m) = 1}
jacobiDn : (F, F) -> F
++ jacobiDn(z, m) is the Jacobi elliptic dn function, defined
++ by \spad{jacobiDn(z, m)^2 + m*jacobiSn(z, m)^2 = 1} and
++ \spad{jacobiDn(0, m) = 1}
jacobiZeta : (F, F) -> F
++ jacobiZeta(z, m) is the Jacobi elliptic zeta function, defined
++ by \spad{D(jacobiZeta(z, m), z) =
++ jacobiDn(z, m)^2 - ellipticE(m)/ellipticK(m)} and
++ \spad{jacobiZeta(0, m) = 0}.
jacobiTheta : (F, F) -> F
++ jacobiTheta(q, z) is the third Jacobi Theta function
lerchPhi : (F, F, F) -> F
++ lerchPhi(z, s, a) is the Lerch Phi function
riemannZeta : F -> F
++ riemannZeta(z) is the Riemann Zeta function
charlierC : (F, F, F) -> F
++ charlierC(n, a, z) is the Charlier polynomial
hermiteH : (F, F) -> F
++ hermiteH(n, z) is the Hermite polynomial
jacobiP : (F, F, F, F) -> F
++ jacobiP(n, a, b, z) is the Jacobi polynomial
laguerreL: (F, F, F) -> F
++ laguerreL(n, a, z) is the Laguerre polynomial
meixnerM : (F, F, F, F) -> F
++ meixnerM(n, b, c, z) is the Meixner polynomial
if F has RetractableTo(Integer) then
hypergeometricF : (List F, List F, F) -> F
++ hypergeometricF(la, lb, z) is the generalized hypergeometric
++ function
meijerG : (List F, List F, List F, List F, F) -> F
++ meijerG(la, lb, lc, ld, z) is the meijerG function
-- Functions below should be local but conditional
iiGamma : F -> F
++ iiGamma(x) should be local but conditional;
iiabs : F -> F
++ iiabs(x) should be local but conditional;
iiconjugate: F -> F
++ iiconjugate(x) should be local but conditional;
iiBeta : List F -> F
++ iiBeta(x) should be local but conditional;
iidigamma : F -> F
++ iidigamma(x) should be local but conditional;
iipolygamma : List F -> F
++ iipolygamma(x) should be local but conditional;
iiBesselJ : List F -> F
++ iiBesselJ(x) should be local but conditional;
iiBesselY : List F -> F
++ iiBesselY(x) should be local but conditional;
iiBesselI : List F -> F
++ iiBesselI(x) should be local but conditional;
iiBesselK : List F -> F
++ iiBesselK(x) should be local but conditional;
iiAiryAi : F -> F
++ iiAiryAi(x) should be local but conditional;
iiAiryAiPrime : F -> F
++ iiAiryAiPrime(x) should be local but conditional;
iiAiryBi : F -> F
++ iiAiryBi(x) should be local but conditional;
iiAiryBiPrime : F -> F
++ iiAiryBiPrime(x) should be local but conditional;
iAiryAi : F -> F
++ iAiryAi(x) should be local but conditional;
iAiryAiPrime : F -> F
++ iAiryAiPrime(x) should be local but conditional;
iAiryBi : F -> F
++ iAiryBi(x) should be local but conditional;
iAiryBiPrime : F -> F
++ iAiryBiPrime(x) should be local but conditional;
iiHypergeometricF : List F -> F
++ iiHypergeometricF(l) should be local but conditional;
iiPolylog : (F, F) -> F
++ iiPolylog(x, s) should be local but conditional;
iLambertW : F -> F
++ iLambertW(x) should be local but conditional;
Implementation ==> add
SPECIAL := 'special
INP ==> InputForm
SPECIALINPUT ==> '%specialInput
iabs : F -> F
iGamma : F -> F
iBeta : (F, F) -> F
idigamma : F -> F
iiipolygamma : (F, F) -> F
iiiBesselJ : (F, F) -> F
iiiBesselY : (F, F) -> F
iiiBesselI : (F, F) -> F
iiiBesselK : (F, F) -> F
iPolylog : List F -> F
iWeierstrassP : (F, F, F) -> F
iWeierstrassPPrime : (F, F, F) -> F
iWeierstrassSigma : (F, F, F) -> F
iWeierstrassZeta : (F, F, F) -> F
iiWeierstrassP : List F -> F
iiWeierstrassPPrime : List F -> F
iiWeierstrassSigma : List F -> F
iiWeierstrassZeta : List F -> F
iiMeijerG : List F -> F
opabs := operator('abs)$CommonOperators
opconjugate := operator('conjugate)$CommonOperators
opGamma := operator('Gamma)$CommonOperators
opGamma2 := operator('Gamma2)$CommonOperators
opBeta := operator('Beta)$CommonOperators
opdigamma := operator('digamma)$CommonOperators
oppolygamma := operator('polygamma)$CommonOperators
opBesselJ := operator('besselJ)$CommonOperators
opBesselY := operator('besselY)$CommonOperators
opBesselI := operator('besselI)$CommonOperators
opBesselK := operator('besselK)$CommonOperators
opAiryAi := operator('airyAi)$CommonOperators
opAiryAiPrime := operator('airyAiPrime)$CommonOperators
opAiryBi := operator('airyBi)$CommonOperators
opAiryBiPrime := operator('airyBiPrime)$CommonOperators
opLambertW := operator('lambertW)$CommonOperators
opPolylog := operator('polylog)$CommonOperators
opWeierstrassP := operator('weierstrassP)$CommonOperators
opWeierstrassPPrime := operator('weierstrassPPrime)$CommonOperators
opWeierstrassSigma := operator('weierstrassSigma)$CommonOperators
opWeierstrassZeta := operator('weierstrassZeta)$CommonOperators
opHypergeometricF := operator('hypergeometricF)$CommonOperators
opMeijerG := operator('meijerG)$CommonOperators
opCharlierC := operator('charlierC)$CommonOperators
opHermiteH := operator('hermiteH)$CommonOperators
opJacobiP := operator('jacobiP)$CommonOperators
opLaguerreL := operator('laguerreL)$CommonOperators
opMeixnerM := operator('meixnerM)$CommonOperators
op_log_gamma := operator('%logGamma)$CommonOperators
op_eis := operator('%eis)$CommonOperators
op_erfs := operator('%erfs)$CommonOperators
op_erfis := operator('%erfis)$CommonOperators
abs x == opabs x
conjugate x == opconjugate x
Gamma(x) == opGamma(x)
Gamma(a, x) == opGamma2(a, x)
Beta(x, y) == opBeta(x, y)
digamma x == opdigamma(x)
polygamma(k, x)== oppolygamma(k, x)
besselJ(a, x) == opBesselJ(a, x)
besselY(a, x) == opBesselY(a, x)
besselI(a, x) == opBesselI(a, x)
besselK(a, x) == opBesselK(a, x)
airyAi(x) == opAiryAi(x)
airyAiPrime(x) == opAiryAiPrime(x)
airyBi(x) == opAiryBi(x)
airyBiPrime(x) == opAiryBiPrime(x)
lambertW(x) == opLambertW(x)
polylog(s, x) == opPolylog(s, x)
weierstrassP(g2, g3, x) == opWeierstrassP(g2, g3, x)
weierstrassPPrime(g2, g3, x) == opWeierstrassPPrime(g2, g3, x)
weierstrassSigma(g2, g3, x) == opWeierstrassSigma(g2, g3, x)
weierstrassZeta(g2, g3, x) == opWeierstrassZeta(g2, g3, x)
if F has RetractableTo(Integer) then
hypergeometricF(a, b, z) ==
nai := #a
nbi := #b
z = 0 and nai <= nbi + 1 => 1
p := (#a)::F
q := (#b)::F
opHypergeometricF concat(concat(a, concat(b, [z])), [p, q])
meijerG(a, b, c, d, z) ==
n1 := (#a)::F
n2 := (#b)::F
m1 := (#c)::F
m2 := (#d)::F
opMeijerG concat(concat(a, concat(b,
concat(c, concat(d, [z])))), [n1, n2, m1, m2])
import from List Kernel(F)
opdiff := operator(operator('%diff)$CommonOperators)$F
dummy ==> new()$SE :: F
ahalf : F := recip(2::F)::F
athird : F := recip(3::F)::F
afourth : F := recip(4::F)::F
asixth : F := recip(6::F)::F
twothirds : F := 2athird
threehalfs : F := 3ahalf
-- Helpers for partially defined derivatives
grad2(l : List F, t : SE, op : OP, d2 : (F, F) -> F ) : F ==
x1 := l(1)
x2 := l(2)
dm := dummy
differentiate(x1, t)*kernel(opdiff, [op [dm, x2], dm, x1])
+ differentiate(x2, t)*d2(x1, x2)
grad3(l : List F, t : SE, op : OP, d3 : (F, F, F) -> F ) : F ==
x1 := l(1)
x2 := l(2)
x3 := l(3)
dm1 := dummy
dm2 := dummy
differentiate(x1, t)*kernel(opdiff, [op [dm1, x2, x3], dm1, x1])
+ differentiate(x2, t)*kernel(opdiff, [op [x1, dm2, x3], dm2, x2])
+ differentiate(x3, t)*d3(x1, x2, x3)
grad4(l : List F, t : SE, op : OP, d4 : (F, F, F, F) -> F ) : F ==
x1 := l(1)
x2 := l(2)
x3 := l(3)
x4 := l(4)
dm1 := dummy
dm2 := dummy
dm3 := dummy
kd1 := kernel(opdiff, [op [dm1, x2, x3, x4], dm1, x1])
kd2 := kernel(opdiff, [op [x1, dm2, x3, x4], dm2, x2])
kd3 := kernel(opdiff, [op [x1, x2, dm3, x4], dm3, x3])
differentiate(x1, t)kd1 + differentiate(x2, t)kd2 +
differentiate(x3, t)kd3 +
differentiate(x4, t)d4(x1, x2, x3, x4)
-- handle WeierstrassPInverse
)if false
opWeierstrassPInverse := operator('weierstrassPInverse)$CommonOperators
weierstrassPInverse(g2, g3, z) == opWeierstrassPInverse(g2, g3, z)
eWeierstrassPInverse(g2 : F, g3 : F, z : F) : F ==
kernel(opWeierstrassPInverse, [g2, g3, z])
elWeierstrassPInverse(l : List F) : F == eWeierstrassPInverse(l(1), l(2), l(3))
evaluate(opWeierstrassPInverse, elWeierstrassPInverse)$BasicOperatorFunctions1(F)
if F has RadicalCategory then
eWeierstrassPInverseGrad_g2(l : List F) : F ==
g2 := l(1)
g3 := l(2)
z := l(3)
error "unimplemented"
eWeierstrassPInverseGrad_g3(l : List F) : F ==
g2 := l(1)
g3 := l(2)
z := l(3)
error "unimplemented"
eWeierstrassPInverseGrad_z(l : List F) : F ==
g2 := l(1)
g3 := l(2)
z := l(3)
1/sqrt(4z^3 - g2z - g3)
derivative(opWeierstrassPInverse, [eWeierstrassPInverseGrad_g2,
eWeierstrassPInverseGrad_g3, eWeierstrassPInverseGrad_z])
)endif
-- handle WhittakerM
opWhittakerM := operator('whittakerM)$CommonOperators
whittakerM(k, m, z) == opWhittakerM(k, m, z)
eWhittakerM(k : F, m : F, z : F) : F ==
kernel(opWhittakerM, [k, m, z])
elWhittakerM(l : List F) : F == eWhittakerM(l(1), l(2), l(3))
evaluate(opWhittakerM, elWhittakerM)$BasicOperatorFunctions1(F)
eWhittakerMGrad_z(k : F, m : F, z : F) : F ==
(ahalf - k/z)whittakerM(k, m, z) +
(ahalf + k + m)whittakerM(k + 1, m, z)/z
dWhittakerM(l : List F, t : SE) : F ==
grad3(l, t, opWhittakerM, eWhittakerMGrad_z)
setProperty(opWhittakerM, SPECIALDIFF, dWhittakerM@((List F, SE)->F)
pretend None)
-- handle WhittakerW
opWhittakerW := operator('whittakerW)$CommonOperators
whittakerW(k, m, z) == opWhittakerW(k, m, z)
eWhittakerW(k : F, m : F, z : F) : F ==
kernel(opWhittakerW, [k, m, z])
elWhittakerW(l : List F) : F == eWhittakerW(l(1), l(2), l(3))
evaluate(opWhittakerW, elWhittakerW)$BasicOperatorFunctions1(F)
eWhittakerWGrad_z(k : F, m : F, z : F) : F ==
(ahalf - k/z)*whittakerW(k, m, z) - whittakerW(k + 1, m, z)/z
dWhittakerW(l : List F, t : SE) : F ==
grad3(l, t, opWhittakerW, eWhittakerWGrad_z)
setProperty(opWhittakerW, SPECIALDIFF, dWhittakerW@((List F, SE)->F)
pretend None)
-- handle AngerJ
opAngerJ := operator('angerJ)$CommonOperators
angerJ(v, z) == opAngerJ(v, z)
if F has TranscendentalFunctionCategory then
eAngerJ(v : F, z : F) : F ==
z = 0 => sin(vpi())/(vpi())
kernel(opAngerJ, [v, z])
elAngerJ(l : List F) : F == eAngerJ(l(1), l(2))
evaluate(opAngerJ, elAngerJ)$BasicOperatorFunctions1(F)
eAngerJGrad_z(v : F, z : F) : F ==
-angerJ(v + 1, z) + vangerJ(v, z)/z - sin(vpi())/(pi()*z)
dAngerJ(l : List F, t : SE) : F ==
grad2(l, t, opAngerJ, eAngerJGrad_z)
setProperty(opAngerJ, SPECIALDIFF, dAngerJ@((List F, SE)->F)
pretend None)
else
eeAngerJ(l : List F) : F == kernel(opAngerJ, l)
evaluate(opAngerJ, eeAngerJ)$BasicOperatorFunctions1(F)
-- handle WeberE
opWeberE := operator('weberE)$CommonOperators
weberE(v, z) == opWeberE(v, z)
if F has TranscendentalFunctionCategory then
eWeberE(v : F, z : F) : F ==
z = 0 => 2sin(ahalfvpi())^2/(vpi())
kernel(opWeberE, [v, z])
elWeberE(l : List F) : F == eWeberE(l(1), l(2))
evaluate(opWeberE, elWeberE)$BasicOperatorFunctions1(F)
eWeberEGrad_z(v : F, z : F) : F ==
-weberE(v + 1, z) + vweberE(v, z)/z - (1 - cos(vpi()))/(pi()*z)
dWeberE(l : List F, t : SE) : F ==
grad2(l, t, opWeberE, eWeberEGrad_z)
setProperty(opWeberE, SPECIALDIFF, dWeberE@((List F, SE)->F)
pretend None)
else
eeWeberE(l : List F) : F == kernel(opWeberE, l)
evaluate(opWeberE, eeWeberE)$BasicOperatorFunctions1(F)
-- handle StruveH
opStruveH := operator('struveH)$CommonOperators
struveH(v, z) == opStruveH(v, z)
eStruveH(v : F, z : F) : F ==
kernel(opStruveH, [v, z])
elStruveH(l : List F) : F == eStruveH(l(1), l(2))
evaluate(opStruveH, elStruveH)$BasicOperatorFunctions1(F)
if F has TranscendentalFunctionCategory
and F has RadicalCategory then
eStruveHGrad_z(v : F, z : F) : F ==
-struveH(v + 1, z) + vstruveH(v, z)/z +
(ahalfz)^v/(sqrt(pi())*Gamma(v + threehalfs))
dStruveH(l : List F, t : SE) : F ==
grad2(l, t, opStruveH, eStruveHGrad_z)
setProperty(opStruveH, SPECIALDIFF, dStruveH@((List F, SE)->F)
pretend None)
-- handle StruveL
opStruveL := operator('struveL)$CommonOperators
struveL(v, z) == opStruveL(v, z)
eStruveL(v : F, z : F) : F ==
kernel(opStruveL, [v, z])
elStruveL(l : List F) : F == eStruveL(l(1), l(2))
evaluate(opStruveL, elStruveL)$BasicOperatorFunctions1(F)
if F has TranscendentalFunctionCategory
and F has RadicalCategory then
eStruveLGrad_z(v : F, z : F) : F ==
struveL(v + 1, z) + vstruveL(v, z)/z +
(ahalfz)^v/(sqrt(pi())*Gamma(v + threehalfs))
dStruveL(l : List F, t : SE) : F ==
grad2(l, t, opStruveL, eStruveLGrad_z)
setProperty(opStruveL, SPECIALDIFF, dStruveL@((List F, SE)->F)
pretend None)
-- handle HankelH1
opHankelH1 := operator('hankelH1)$CommonOperators
hankelH1(v, z) == opHankelH1(v, z)
eHankelH1(v : F, z : F) : F ==
kernel(opHankelH1, [v, z])
elHankelH1(l : List F) : F == eHankelH1(l(1), l(2))
evaluate(opHankelH1, elHankelH1)$BasicOperatorFunctions1(F)
eHankelH1Grad_z(v : F, z : F) : F ==
-hankelH1(v + 1, z) + v*hankelH1(v, z)/z
dHankelH1(l : List F, t : SE) : F ==
grad2(l, t, opHankelH1, eHankelH1Grad_z)
setProperty(opHankelH1, SPECIALDIFF, dHankelH1@((List F, SE)->F)
pretend None)
-- handle HankelH2
opHankelH2 := operator('hankelH2)$CommonOperators
hankelH2(v, z) == opHankelH2(v, z)
eHankelH2(v : F, z : F) : F ==
kernel(opHankelH2, [v, z])
elHankelH2(l : List F) : F == eHankelH2(l(1), l(2))
evaluate(opHankelH2, elHankelH2)$BasicOperatorFunctions1(F)
eHankelH2Grad_z(v : F, z : F) : F ==
-hankelH2(v + 1, z) + v*hankelH2(v, z)/z
dHankelH2(l : List F, t : SE) : F ==
grad2(l, t, opHankelH2, eHankelH2Grad_z)
setProperty(opHankelH2, SPECIALDIFF, dHankelH2@((List F, SE)->F)
pretend None)
-- handle LommelS1
opLommelS1 := operator('lommelS1)$CommonOperators
lommelS1(m, v, z) == opLommelS1(m, v, z)
eLommelS1(m : F, v : F, z : F) : F ==
kernel(opLommelS1, [m, v, z])
elLommelS1(l : List F) : F == eLommelS1(l(1), l(2), l(3))
evaluate(opLommelS1, elLommelS1)$BasicOperatorFunctions1(F)
eLommelS1Grad_z(m : F, v : F, z : F) : F ==
-vlommelS1(m, v, z)/z + (m + v - 1)lommelS1(m - 1, v - 1, z)
dLommelS1(l : List F, t : SE) : F ==
grad3(l, t, opLommelS1, eLommelS1Grad_z)
setProperty(opLommelS1, SPECIALDIFF, dLommelS1@((List F, SE)->F)
pretend None)
-- handle LommelS2
opLommelS2 := operator('lommelS2)$CommonOperators
lommelS2(mu, nu, z) == opLommelS2(mu, nu, z)
eLommelS2(mu : F, nu : F, z : F) : F ==
kernel(opLommelS2, [mu, nu, z])
elLommelS2(l : List F) : F == eLommelS2(l(1), l(2), l(3))
evaluate(opLommelS2, elLommelS2)$BasicOperatorFunctions1(F)
eLommelS2Grad_z(m : F, v : F, z : F) : F ==
-vlommelS2(m, v, z)/z + (m + v - 1)lommelS2(m - 1, v - 1, z)
dLommelS2(l : List F, t : SE) : F ==
grad3(l, t, opLommelS2, eLommelS2Grad_z)
setProperty(opLommelS2, SPECIALDIFF, dLommelS2@((List F, SE)->F)
pretend None)
-- handle KummerM
opKummerM := operator('kummerM)$CommonOperators
kummerM(mu, nu, z) == opKummerM(mu, nu, z)
eKummerM(a : F, b : F, z : F) : F ==
z = 0 => 1
kernel(opKummerM, [a, b, z])
elKummerM(l : List F) : F == eKummerM(l(1), l(2), l(3))
evaluate(opKummerM, elKummerM)$BasicOperatorFunctions1(F)
eKummerMGrad_z(a : F, b : F, z : F) : F ==
((z + a - b)kummerM(a, b, z)+(b - a)kummerM(a - 1, b, z))/z
dKummerM(l : List F, t : SE) : F ==
grad3(l, t, opKummerM, eKummerMGrad_z)
setProperty(opKummerM, SPECIALDIFF, dKummerM@((List F, SE)->F)
pretend None)
-- handle KummerU
opKummerU := operator('kummerU)$CommonOperators
kummerU(a, b, z) == opKummerU(a, b, z)
eKummerU(a : F, b : F, z : F) : F ==
kernel(opKummerU, [a, b, z])
elKummerU(l : List F) : F == eKummerU(l(1), l(2), l(3))
evaluate(opKummerU, elKummerU)$BasicOperatorFunctions1(F)
eKummerUGrad_z(a : F, b : F, z : F) : F ==
((z + a - b)*kummerU(a, b, z) - kummerU(a - 1, b, z))/z
dKummerU(l : List F, t : SE) : F ==
grad3(l, t, opKummerU, eKummerUGrad_z)
setProperty(opKummerU, SPECIALDIFF, dKummerU@((List F, SE)->F)
pretend None)
-- handle LegendreP
opLegendreP := operator('legendreP)$CommonOperators
legendreP(nu, mu, z) == opLegendreP(nu, mu, z)
eLegendreP(nu : F, mu : F, z : F) : F ==
kernel(opLegendreP, [nu, mu, z])
elLegendreP(l : List F) : F == eLegendreP(l(1), l(2), l(3))
evaluate(opLegendreP, elLegendreP)$BasicOperatorFunctions1(F)
eLegendrePGrad_z(nu : F, mu : F, z : F) : F ==
(nu - mu + 1)legendreP(nu + 1, mu, z) -
(nu + 1)z*legendreP(nu, mu, z)
dLegendreP(l : List F, t : SE) : F ==
grad3(l, t, opLegendreP, eLegendrePGrad_z)
setProperty(opLegendreP, SPECIALDIFF, dLegendreP@((List F, SE)->F)
pretend None)
-- handle LegendreQ
opLegendreQ := operator('legendreQ)$CommonOperators
legendreQ(nu, mu, z) == opLegendreQ(nu, mu, z)
eLegendreQ(nu : F, mu : F, z : F) : F ==
kernel(opLegendreQ, [nu, mu, z])
elLegendreQ(l : List F) : F == eLegendreQ(l(1), l(2), l(3))
evaluate(opLegendreQ, elLegendreQ)$BasicOperatorFunctions1(F)
eLegendreQGrad_z(nu : F, mu : F, z : F) : F ==
(nu - mu + 1)legendreQ(nu + 1, mu, z) -
(nu + 1)z*legendreQ(nu, mu, z)
dLegendreQ(l : List F, t : SE) : F ==
grad3(l, t, opLegendreQ, eLegendreQGrad_z)
setProperty(opLegendreQ, SPECIALDIFF, dLegendreQ@((List F, SE)->F)
pretend None)
-- handle KelvinBei
opKelvinBei := operator('kelvinBei)$CommonOperators
kelvinBei(v, z) == opKelvinBei(v, z)
eKelvinBei(v : F, z : F) : F ==
kernel(opKelvinBei, [v, z])
elKelvinBei(l : List F) : F == eKelvinBei(l(1), l(2))
evaluate(opKelvinBei, elKelvinBei)$BasicOperatorFunctions1(F)
if F has RadicalCategory then
eKelvinBeiGrad_z(v : F, z : F) : F ==
ahalfsqrt(2::F)(kelvinBei(v + 1, z) - kelvinBer(v + 1, z)) +
v*kelvinBei(v, z)/z
dKelvinBei(l : List F, t : SE) : F ==
grad2(l, t, opKelvinBei, eKelvinBeiGrad_z)
setProperty(opKelvinBei, SPECIALDIFF, dKelvinBei@((List F, SE)->F)
pretend None)
-- handle KelvinBer
opKelvinBer := operator('kelvinBer)$CommonOperators
kelvinBer(v, z) == opKelvinBer(v, z)
eKelvinBer(v : F, z : F) : F ==
kernel(opKelvinBer, [v, z])
elKelvinBer(l : List F) : F == eKelvinBer(l(1), l(2))
evaluate(opKelvinBer, elKelvinBer)$BasicOperatorFunctions1(F)
if F has RadicalCategory then
eKelvinBerGrad_z(v : F, z : F) : F ==
ahalfsqrt(2::F)(kelvinBer(v + 1, z) + kelvinBei(v + 1, z)) +
v*kelvinBer(v, z)/z
dKelvinBer(l : List F, t : SE) : F ==
grad2(l, t, opKelvinBer, eKelvinBerGrad_z)
setProperty(opKelvinBer, SPECIALDIFF, dKelvinBer@((List F, SE)->F)
pretend None)
-- handle KelvinKei
opKelvinKei := operator('kelvinKei)$CommonOperators
kelvinKei(v, z) == opKelvinKei(v, z)
eKelvinKei(v : F, z : F) : F ==
kernel(opKelvinKei, [v, z])
elKelvinKei(l : List F) : F == eKelvinKei(l(1), l(2))
evaluate(opKelvinKei, elKelvinKei)$BasicOperatorFunctions1(F)
if F has RadicalCategory then
eKelvinKeiGrad_z(v : F, z : F) : F ==
ahalfsqrt(2::F)(kelvinKei(v + 1, z) - kelvinKer(v + 1, z)) +
v*kelvinKei(v, z)/z
dKelvinKei(l : List F, t : SE) : F ==
grad2(l, t, opKelvinKei, eKelvinKeiGrad_z)
setProperty(opKelvinKei, SPECIALDIFF, dKelvinKei@((List F, SE)->F)
pretend None)
-- handle KelvinKer
opKelvinKer := operator('kelvinKer)$CommonOperators
kelvinKer(v, z) == opKelvinKer(v, z)
eKelvinKer(v : F, z : F) : F ==
kernel(opKelvinKer, [v, z])
elKelvinKer(l : List F) : F == eKelvinKer(l(1), l(2))
evaluate(opKelvinKer, elKelvinKer)$BasicOperatorFunctions1(F)
if F has RadicalCategory then
eKelvinKerGrad_z(v : F, z : F) : F ==
ahalfsqrt(2::F)(kelvinKer(v + 1, z) + kelvinKei(v + 1, z)) +
v*kelvinKer(v, z)/z
dKelvinKer(l : List F, t : SE) : F ==
grad2(l, t, opKelvinKer, eKelvinKerGrad_z)
setProperty(opKelvinKer, SPECIALDIFF, dKelvinKer@((List F, SE)->F)
pretend None)
-- handle EllipticK
opEllipticK := operator('ellipticK)$CommonOperators
ellipticK(m) == opEllipticK(m)
eEllipticK(m : F) : F ==
kernel(opEllipticK, [m])
elEllipticK(l : List F) : F == eEllipticK(l(1))
evaluate(opEllipticK, elEllipticK)$BasicOperatorFunctions1(F)
dEllipticK(m : F) : F ==
ahalf(ellipticE(m) - (1 - m)ellipticK(m))/(m*(1 - m))
derivative(opEllipticK, dEllipticK)
-- handle one argument EllipticE
opEllipticE := operator('ellipticE)$CommonOperators
ellipticE(m) == opEllipticE(m)
eEllipticE(m : F) : F ==
kernel(opEllipticE, [m])
elEllipticE(l : List F) : F == eEllipticE(l(1))
evaluate(opEllipticE, elEllipticE)$BasicOperatorFunctions1(F)
dEllipticE(m : F) : F ==
ahalf*(ellipticE(m) - ellipticK(m))/m
derivative(opEllipticE, dEllipticE)
-- handle two argument EllipticE
opEllipticE2 := operator('ellipticE2)$CommonOperators
ellipticE(z, m) == opEllipticE2(z, m)
eEllipticE2(z : F, m : F) : F ==
z = 0 => 0
z = 1 => eEllipticE(m)
kernel(opEllipticE2, [z, m])
elEllipticE2(l : List F) : F == eEllipticE2(l(1), l(2))
evaluate(opEllipticE2, elEllipticE2)$BasicOperatorFunctions1(F)
if F has RadicalCategory then
eEllipticE2Grad_z(l : List F) : F ==
z := l(1)
m := l(2)
sqrt(1 - m*z^2)/sqrt(1 - z^2)
eEllipticE2Grad_m(l : List F) : F ==
z := l(1)
m := l(2)
ahalf*(ellipticE(z, m) - ellipticF(z, m))/m
derivative(opEllipticE2, [eEllipticE2Grad_z, eEllipticE2Grad_m])
inEllipticE2(li : List INP) : INP ==
convert cons(convert('ellipticE), li)
input(opEllipticE2, inEllipticE2@((List INP) -> INP))
-- handle EllipticF
opEllipticF := operator('ellipticF)$CommonOperators
ellipticF(z, m) == opEllipticF(z, m)
eEllipticF(z : F, m : F) : F ==
z = 0 => 0
z = 1 => ellipticK(m)
kernel(opEllipticF, [z, m])
elEllipticF(l : List F) : F == eEllipticF(l(1), l(2))
evaluate(opEllipticF, elEllipticF)$BasicOperatorFunctions1(F)
if F has RadicalCategory then
eEllipticFGrad_z(l : List F) : F ==
z := l(1)
m := l(2)
1/(sqrt(1 - mz^2)sqrt(1 - z^2))
eEllipticFGrad_m(l : List F) : F ==
z := l(1)
m := l(2)
ahalf((ellipticE(z, m) - (1 - m)ellipticF(z, m))/m -
zsqrt(1 - z^2)/sqrt(1 - mz^2))/(1 - m)
derivative(opEllipticF, [eEllipticFGrad_z, eEllipticFGrad_m])
-- handle EllipticPi
opEllipticPi := operator('ellipticPi)$CommonOperators
ellipticPi(z, n, m) == opEllipticPi(z, n, m)
eEllipticPi(z : F, n : F, m : F) : F ==
z = 0 => 0
kernel(opEllipticPi, [z, n, m])
elEllipticPi(l : List F) : F == eEllipticPi(l(1), l(2), l(3))
evaluate(opEllipticPi, elEllipticPi)$BasicOperatorFunctions1(F)
if F has RadicalCategory then
eEllipticPiGrad_z(l : List F) : F ==
z := l(1)
n := l(2)
m := l(3)
1/((1 - nz^2)sqrt(1 - mz^2)sqrt(1 - z^2))
eEllipticPiGrad_n(l : List F) : F ==
z := l(1)
n := l(2)
m := l(3)
t1 := -(n^2 - m)ellipticPi(z, n, m)/((n - 1)(n - m)n)
t2 := ellipticF(z, m)/((n - 1)n)
t3 := -ellipticE(z, m)/((n - 1)(n - m))
t4 := nzsqrt(1 - mz^2)sqrt(1 - z^2)/
((1 - nz^2)(n - 1)(n - m))
ahalf*(t1 + t2 + t3 + t4)
eEllipticPiGrad_m(l : List F) : F ==
z := l(1)
n := l(2)
m := l(3)
t1 := mzsqrt(1 - z^2)/sqrt(1 - mz^2)
t2 := (-ellipticE(z, m) + t1)/(1 - m)
ahalf(ellipticPi(z, n, m) + t2)/(n - m)
derivative(opEllipticPi, [eEllipticPiGrad_z, eEllipticPiGrad_n,
eEllipticPiGrad_m])
-- handle JacobiSn
opJacobiSn := operator('jacobiSn)$CommonOperators
jacobiSn(z, m) == opJacobiSn(z, m)
eJacobiSn(z : F, m : F) : F ==
z = 0 => 0
if is?(z, opEllipticF) then
args := argument(retract(z)@K)
m = args(2) => return args(1)
kernel(opJacobiSn, [z, m])
elJacobiSn : List F -> F
elJacobiSn(l : List F) : F == eJacobiSn(l(1), l(2))
evaluate(opJacobiSn, elJacobiSn)$BasicOperatorFunctions1(F)
jacobiGradHelper(z : F, m : F) : F ==
(z - ellipticE(jacobiSn(z, m), m)/(1 - m))/m
eJacobiSnGrad_z(l : List F) : F ==
z := l(1)
m := l(2)
jacobiCn(z, m)*jacobiDn(z, m)
eJacobiSnGrad_m(l : List F) : F ==
z := l(1)
m := l(2)
ahalf(eJacobiSnGrad_z(l)jacobiGradHelper(z, m) +
jacobiSn(z, m)*jacobiCn(z, m)^2/(1 - m))
derivative(opJacobiSn, [eJacobiSnGrad_z, eJacobiSnGrad_m])
-- handle JacobiCn
opJacobiCn := operator('jacobiCn)$CommonOperators
jacobiCn(z, m) == opJacobiCn(z, m)
eJacobiCn(z : F, m : F) : F ==
z = 0 => 1
kernel(opJacobiCn, [z, m])
elJacobiCn(l : List F) : F == eJacobiCn(l(1), l(2))
evaluate(opJacobiCn, elJacobiCn)$BasicOperatorFunctions1(F)
eJacobiCnGrad_z(l : List F) : F ==
z := l(1)
m := l(2)
-jacobiSn(z, m)*jacobiDn(z, m)
eJacobiCnGrad_m(l : List F) : F ==
z := l(1)
m := l(2)
ahalf(eJacobiCnGrad_z(l)jacobiGradHelper(z, m) -
jacobiSn(z, m)^2*jacobiCn(z, m)/(1 - m))
derivative(opJacobiCn, [eJacobiCnGrad_z, eJacobiCnGrad_m])
-- handle JacobiDn
opJacobiDn := operator('jacobiDn)$CommonOperators
jacobiDn(z, m) == opJacobiDn(z, m)
eJacobiDn(z : F, m : F) : F ==
z = 0 => 1
kernel(opJacobiDn, [z, m])
elJacobiDn(l : List F) : F == eJacobiDn(l(1), l(2))
evaluate(opJacobiDn, elJacobiDn)$BasicOperatorFunctions1(F)
eJacobiDnGrad_z(l : List F) : F ==
z := l(1)
m := l(2)
-mjacobiSn(z, m)jacobiCn(z, m)
eJacobiDnGrad_m(l : List F) : F ==
z := l(1)
m := l(2)
ahalf(eJacobiDnGrad_z(l)jacobiGradHelper(z, m) -
jacobiSn(z, m)^2*jacobiDn(z, m)/(1 - m))
derivative(opJacobiDn, [eJacobiDnGrad_z, eJacobiDnGrad_m])
-- handle JacobiZeta
opJacobiZeta := operator('jacobiZeta)$CommonOperators
jacobiZeta(z, m) == opJacobiZeta(z, m)
eJacobiZeta(z : F, m : F) : F ==
z = 0 => 0
kernel(opJacobiZeta, [z, m])
elJacobiZeta(l : List F) : F == eJacobiZeta(l(1), l(2))
evaluate(opJacobiZeta, elJacobiZeta)$BasicOperatorFunctions1(F)
eJacobiZetaGrad_z(l : List F) : F ==
z := l(1)
m := l(2)
dn := jacobiDn(z, m)
dn*dn - ellipticE(m)/ellipticK(m)
eJacobiZetaGrad_m(l : List F) : F ==
z := l(1)
m := l(2)
ek := ellipticK(m)
ee := ellipticE(m)
er := ee/ek
dn := jacobiDn(z, m)
res1 := (dndn + m - 1)jacobiZeta(z, m)
res2 := res1 + (m - 1)zdndn
res3 := res2 - mjacobiCn(z, m)jacobiDn(z, m)jacobiSn(z, m)
res4 := res3 + z(1 - m + dndn)er
ahalf(res4 - zerer)/(m*m - m)
derivative(opJacobiZeta, [eJacobiZetaGrad_z, eJacobiZetaGrad_m])
-- handle JacobiTheta
opJacobiTheta := operator('jacobiTheta)$CommonOperators
jacobiTheta(q, z) == opJacobiTheta(q, z)
eJacobiTheta(q : F, z : F) : F ==
kernel(opJacobiTheta, [q, z])
elJacobiTheta(l : List F) : F == eJacobiTheta(l(1), l(2))
evaluate(opJacobiTheta, elJacobiTheta)$BasicOperatorFunctions1(F)
-- handle LerchPhi
opLerchPhi := operator('lerchPhi)$CommonOperators
lerchPhi(z, s, a) == opLerchPhi(z, s, a)
- eLerchPhi(z : F, s : F, a : F) : F ==
- z = 0 => 1/a^s
a = 1 => polylog(s, z)/z
kernel(opLerchPhi, [z, s, a])
elLerchPhi(l : List F) : F == eLerchPhi(l(1), l(2), l(3))
evaluate(opLerchPhi, elLerchPhi)$BasicOperatorFunctions1(F)
dLerchPhi(l : List F, t : SE) : F ==
z := l(1)
s := l(2)
a := l(3)
dz := differentiate(z, t)(lerchPhi(z, s - 1, a) -
alerchPhi(z, s, a))/z
da := -differentiate(a, t)slerchPhi(z, s + 1, a)
dm := dummy
differentiate(s, t)*kernel(opdiff, [opLerchPhi [z, dm, a], dm, s])
+ dz + da
setProperty(opLerchPhi, SPECIALDIFF, dLerchPhi@((List F, SE)->F)
pretend None)
-- handle RiemannZeta
opRiemannZeta := operator('riemannZeta)$CommonOperators
riemannZeta(z) == opRiemannZeta(z)
eRiemannZeta(z : F) : F ==
kernel(opRiemannZeta, [z])
elRiemannZeta(l : List F) : F == eRiemannZeta(l(1))
evaluate(opRiemannZeta, elRiemannZeta)$BasicOperatorFunctions1(F)
-- orthogonal polynomials
charlierC(n : F, a : F, z : F) : F == opCharlierC(n, a, z)
eCharlierC(n : F, a : F, z : F) : F ==
n = 0 => 1
n = 1 => (z - a)/a
kernel(opCharlierC, [n, a, z])
elCharlierC(l : List F) : F == eCharlierC(l(1), l(2), l(3))
evaluate(opCharlierC, elCharlierC)$BasicOperatorFunctions1(F)
hermiteH(n : F, z: F) : F == opHermiteH(n, z)
eHermiteH(n : F, z: F) : F ==
n = -1 => 0
n = 0 => 1
n = 1 => (2::F)*z
kernel(opHermiteH, [n, z])
elHermiteH(l : List F) : F == eHermiteH(l(1), l(2))
evaluate(opHermiteH, elHermiteH)$BasicOperatorFunctions1(F)
eHermiteHGrad_z(n : F, z : F) : F == (2::F)nhermiteH(n - 1, z)
dHermiteH(l : List F, t : SE) : F ==
grad2(l, t, opHermiteH, eHermiteHGrad_z)
setProperty(opHermiteH, SPECIALDIFF, dHermiteH@((List F, SE)->F)
pretend None)
jacobiP(n : F, a : F, b : F, z : F) : F == opJacobiP(n, a, b, z)
eJacobiP(n : F, a : F, b : F, z : F) : F ==
n = -1 => 0
n = 0 => 1
n = 1 => ahalf(a - b) + (1 + ahalf(a + b))*z
kernel(opJacobiP, [n, a, b, z])
elJacobiP(l : List F) : F == eJacobiP(l(1), l(2), l(3), l(4))
evaluate(opJacobiP, elJacobiP)$BasicOperatorFunctions1(F)
eJacobiPGrad_z(n : F, a : F, b : F, z : F) : F ==
ahalf(a + b + n + 1)jacobiP(n - 1, a + 1, b + 1, z)
dJacobiP(l : List F, t : SE) : F ==
grad4(l, t, opJacobiP, eJacobiPGrad_z)
setProperty(opJacobiP, SPECIALDIFF, dJacobiP@((List F, SE)->F)
pretend None)
laguerreL(n : F, a : F, z : F) : F == opLaguerreL(n, a, z)
eLaguerreL(n : F, a : F, z : F) : F ==
n = -1 => 0
n = 0 => 1
n = 1 => (1 + a - z)
kernel(opLaguerreL, [n, a, z])
elLaguerreL(l : List F) : F == eLaguerreL(l(1), l(2), l(3))
evaluate(opLaguerreL, elLaguerreL)$BasicOperatorFunctions1(F)
eLaguerreLGrad_z(n : F, a : F, z : F) : F ==
laguerreL(n - 1, a + 1, z)
dLaguerreL(l : List F, t : SE) : F ==
grad3(l, t, opLaguerreL, eLaguerreLGrad_z)
setProperty(opLaguerreL, SPECIALDIFF, dLaguerreL@((List F, SE)->F)
pretend None)
meixnerM(n : F, b : F, c : F, z : F) : F == opMeixnerM(n, b, c, z)
eMeixnerM(n : F, b : F, c : F, z : F) : F ==
n = 0 => 1
n = 1 => (c - 1)z/(cb) + 1
kernel(opMeixnerM, [n, b, c, z])
elMeixnerM(l : List F) : F == eMeixnerM(l(1), l(2), l(3), l(4))
evaluate(opMeixnerM, elMeixnerM)$BasicOperatorFunctions1(F)
--
belong? op == has?(op, SPECIAL)
operator op ==
is?(op, 'abs) => opabs
is?(op, 'conjugate)=> opconjugate
is?(op, 'Gamma) => opGamma
is?(op, 'Gamma2) => opGamma2
is?(op, 'Beta) => opBeta
is?(op, 'digamma) => opdigamma
is?(op, 'polygamma)=> oppolygamma
is?(op, 'besselJ) => opBesselJ
is?(op, 'besselY) => opBesselY
is?(op, 'besselI) => opBesselI
is?(op, 'besselK) => opBesselK
is?(op, 'airyAi) => opAiryAi
is?(op, 'airyAiPrime) => opAiryAiPrime
is?(op, 'airyBi) => opAiryBi
is?(op, 'airyBiPrime) => opAiryBiPrime
is?(op, 'lambertW) => opLambertW
is?(op, 'polylog) => opPolylog
is?(op, 'weierstrassP) => opWeierstrassP
is?(op, 'weierstrassPPrime) => opWeierstrassPPrime
is?(op, 'weierstrassSigma) => opWeierstrassSigma
is?(op, 'weierstrassZeta) => opWeierstrassZeta
is?(op, 'hypergeometricF) => opHypergeometricF
is?(op, 'meijerG) => opMeijerG
-- is?(op, 'weierstrassPInverse) => opWeierstrassPInverse
is?(op, 'whittakerM) => opWhittakerM
is?(op, 'whittakerW) => opWhittakerW
is?(op, 'angerJ) => opAngerJ
is?(op, 'weberE) => opWeberE
is?(op, 'struveH) => opStruveH
is?(op, 'struveL) => opStruveL
is?(op, 'hankelH1) => opHankelH1
is?(op, 'hankelH2) => opHankelH2
is?(op, 'lommelS1) => opLommelS1
is?(op, 'lommelS2) => opLommelS2
is?(op, 'kummerM) => opKummerM
is?(op, 'kummerU) => opKummerU
is?(op, 'legendreP) => opLegendreP
is?(op, 'legendreQ) => opLegendreQ
is?(op, 'kelvinBei) => opKelvinBei
is?(op, 'kelvinBer) => opKelvinBer
is?(op, 'kelvinKei) => opKelvinKei
is?(op, 'kelvinKer) => opKelvinKer
is?(op, 'ellipticK) => opEllipticK
is?(op, 'ellipticE) => opEllipticE
is?(op, 'ellipticE2) => opEllipticE2
is?(op, 'ellipticF) => opEllipticF
is?(op, 'ellipticPi) => opEllipticPi
is?(op, 'jacobiSn) => opJacobiSn
is?(op, 'jacobiCn) => opJacobiCn
is?(op, 'jacobiDn) => opJacobiDn
is?(op, 'jacobiZeta) => opJacobiZeta
is?(op, 'jacobiTheta) => opJacobiTheta
is?(op, 'lerchPhi) => opLerchPhi
is?(op, 'riemannZeta) => opRiemannZeta
is?(op, 'charlierC) => opCharlierC
is?(op, 'hermiteH) => opHermiteH
is?(op, 'jacobiP) => opJacobiP
is?(op, 'laguerreL) => opLaguerreL
is?(op, 'meixnerM) => opMeixnerM
is?(op, '%logGamma) => op_log_gamma
is?(op, '%eis) => op_eis
is?(op, '%erfs) => op_erfs
is?(op, '%erfis) => op_erfis
error "Not a special operator"
-- Could put more unconditional special rules for other functions here
iGamma x ==
-- one? x => x
(x = 1) => x
kernel(opGamma, x)
iabs x ==
zero? x => 0
one? x => 1
is?(x, opabs) => x
is?(x, opconjugate) => kernel(opabs, argument(retract(x)@K)(1))
smaller?(x, 0) => kernel(opabs, -x)
kernel(opabs, x)
iconjugate(k:K):F ==
--output("in: ",k::OutputForm)$OutputPackage
if is?(k, opconjugate) then
x:=argument(k)(1)
else if is?(k, opabs) then
x:= k::F
else if symbolIfCan(k) case Symbol then
x:=kernel(opconjugate, k::F)
else if F has RadicalCategory and R has RetractableTo(Integer) and is?(k, 'nthRoot) then
x:= 1/nthRoot(1/iiconjugate argument(k)(1),retract(argument(k)(2))@Integer)
else -- assume holomorphic
x:=map(iiconjugate,k)
--output("out: ",x::OutputForm)$OutputPackage
return x
iiconjugate(x:F):F ==
--output("iin: ",x::OutputForm)$OutputPackage
if (s:=isPlus(x)) case List F then
--output("isPlus: ",s::OutputForm)$OutputPackage
x := reduce(_+$F,map(iiconjugate,s))
else if (s:=isTimes(x)) case List F then
--output("isTimes: ",s::OutputForm)$OutputPackage
x:= reduce(_*$F,map(iiconjugate,s))
else if #(ks:List K:=kernels(x))>0 then
--output("kernels: ",ks::OutputForm)$OutputPackage
x := eval(x,ks, _
map((k:Kernel F):F +-> (height(k)=0 =>k::F;iconjugate k), _
ks)$ListFunctions2(Kernel F,F))
else if R has conjugate : R -> R then
x:=map(conjugate$R,numer x)::F / _
map(conjugate$R,denom x)::F
--output("iout: ",x::OutputForm)$OutputPackage
return x
iBeta(x, y) == kernel(opBeta, [x, y])
idigamma x == kernel(opdigamma, x)
iiipolygamma(n, x) == kernel(oppolygamma, [n, x])
iiiBesselJ(x, y) == kernel(opBesselJ, [x, y])
iiiBesselY(x, y) == kernel(opBesselY, [x, y])
iiiBesselI(x, y) == kernel(opBesselI, [x, y])
iiiBesselK(x, y) == kernel(opBesselK, [x, y])
import from Fraction(Integer)
if F has ElementaryFunctionCategory then
iAiryAi x ==
zero?(x) => 1::F/((3::F)^twothirdsGamma(twothirds))
kernel(opAiryAi, x)
iAiryAiPrime x ==
zero?(x) => -1::F/((3::F)^athirdGamma(athird))
kernel(opAiryAiPrime, x)
iAiryBi x ==
zero?(x) => 1::F/((3::F)^asixth*Gamma(twothirds))
kernel(opAiryBi, x)
iAiryBiPrime x ==
zero?(x) => (3::F)^asixth/Gamma(athird)
kernel(opAiryBiPrime, x)
else
iAiryAi x == kernel(opAiryAi, x)
iAiryAiPrime x == kernel(opAiryAiPrime, x)
iAiryBi x == kernel(opAiryBi, x)
iAiryBiPrime x == kernel(opAiryBiPrime, x)
if F has ElementaryFunctionCategory then
iLambertW(x) ==
zero?(x) => 0
x = exp(1$F) => 1$F
x = -exp(-1$F) => -1$F
kernel(opLambertW, x)
else
iLambertW(x) ==
zero?(x) => 0
kernel(opLambertW, x)
if F has ElementaryFunctionCategory then
if F has LiouvillianFunctionCategory then
iiPolylog(s, x) ==
s = 1 => -log(1 - x)
s = 2::F => dilog(1 - x)
kernel(opPolylog, [s, x])
else
iiPolylog(s, x) ==
s = 1 => -log(1 - x)
kernel(opPolylog, [s, x])
else
iiPolylog(s, x) == kernel(opPolylog, [s, x])
iPolylog(l) == iiPolylog(first l, second l)
iWeierstrassP(g2, g3, x) == kernel(opWeierstrassP, [g2, g3, x])
iWeierstrassPPrime(g2, g3, x) == kernel(opWeierstrassPPrime, [g2, g3, x])
iWeierstrassSigma(g2, g3, x) ==
x = 0 => 0
kernel(opWeierstrassSigma, [g2, g3, x])
iWeierstrassZeta(g2, g3, x) == kernel(opWeierstrassZeta, [g2, g3, x])
-- Could put more conditional special rules for other functions here
if R has abs : R -> R then
import from Polynomial R
iiabs x ==
(r := retractIfCan(x)@Union(Fraction Polynomial R, "failed"))
case "failed" => iabs x
f := r::Fraction Polynomial R
(a := retractIfCan(numer f)@Union(R, "failed")) case "failed" or
(b := retractIfCan(denom f)@Union(R,"failed")) case "failed" => iabs x
abs(a::R)::F / abs(b::R)::F
else
if F has RadicalCategory and R has conjugate : R -> R then
iiabs x ==
(r := retractIfCan(x)@Union(R, "failed"))
case "failed" => iabs x
sqrt( (r::R*conjugate(r::R))::F)
else iiabs x == iabs x
if R has SpecialFunctionCategory then
iiGamma x ==
(r := retractIfCan(x)@Union(R,"failed")) case "failed" => iGamma x
Gamma(r::R)::F
iiBeta l ==
(r := retractIfCan(first l)@Union(R,"failed")) case "failed" or
(s := retractIfCan(second l)@Union(R,"failed")) case "failed"
=> iBeta(first l, second l)
Beta(r::R, s::R)::F
iidigamma x ==
(r := retractIfCan(x)@Union(R,"failed")) case "failed" => idigamma x
digamma(r::R)::F
iipolygamma l ==
(s := retractIfCan(first l)@Union(R,"failed")) case "failed" or
(r := retractIfCan(second l)@Union(R,"failed")) case "failed"
=> iiipolygamma(first l, second l)
polygamma(s::R, r::R)::F
iiBesselJ l ==
(r := retractIfCan(first l)@Union(R,"failed")) case "failed" or
(s := retractIfCan(second l)@Union(R,"failed")) case "failed"
=> iiiBesselJ(first l, second l)
besselJ(r::R, s::R)::F
iiBesselY l ==
(r := retractIfCan(first l)@Union(R,"failed")) case "failed" or
(s := retractIfCan(second l)@Union(R,"failed")) case "failed"
=> iiiBesselY(first l, second l)
besselY(r::R, s::R)::F
iiBesselI l ==
(r := retractIfCan(first l)@Union(R,"failed")) case "failed" or
(s := retractIfCan(second l)@Union(R,"failed")) case "failed"
=> iiiBesselI(first l, second l)
besselI(r::R, s::R)::F
iiBesselK l ==
(r := retractIfCan(first l)@Union(R,"failed")) case "failed" or
(s := retractIfCan(second l)@Union(R,"failed")) case "failed"
=> iiiBesselK(first l, second l)
besselK(r::R, s::R)::F
iiAiryAi x ==
(r := retractIfCan(x)@Union(R,"failed")) case "failed" => iAiryAi x
airyAi(r::R)::F
iiAiryAiPrime x ==
(r := retractIfCan(x)@Union(R,"failed")) case "failed" => iAiryAiPrime x
airyAiPrime(r::R)::F
iiAiryBi x ==
(r := retractIfCan(x)@Union(R,"failed")) case "failed" => iAiryBi x
airyBi(r::R)::F
iiAiryBi x ==
(r := retractIfCan(x)@Union(R,"failed")) case "failed" => iAiryBiPrime x
airyBiPrime(r::R)::F
else
if R has RetractableTo Integer then
iiGamma x ==
(r := retractIfCan(x)@Union(Integer, "failed")) case Integer
and (r::Integer >= 1) => factorial(r::Integer - 1)::F
iGamma x
else
iiGamma x == iGamma x
iiBeta l == iBeta(first l, second l)
iidigamma x == idigamma x
iipolygamma l == iiipolygamma(first l, second l)
iiBesselJ l == iiiBesselJ(first l, second l)
iiBesselY l == iiiBesselY(first l, second l)
iiBesselI l == iiiBesselI(first l, second l)
iiBesselK l == iiiBesselK(first l, second l)
iiAiryAi x == iAiryAi x
iiAiryAiPrime x == iAiryAiPrime x
iiAiryBi x == iAiryBi x
iiAiryBiPrime x == iAiryBiPrime x
iiWeierstrassP l == iWeierstrassP(first l, second l, third l)
iiWeierstrassPPrime l == iWeierstrassPPrime(first l, second l, third l)
iiWeierstrassSigma l == iWeierstrassSigma(first l, second l, third l)
iiWeierstrassZeta l == iWeierstrassZeta(first l, second l, third l)
-- Default behaviour is to build a kernel
evaluate(opGamma, iiGamma)$BasicOperatorFunctions1(F)
evaluate(opabs, iiabs)$BasicOperatorFunctions1(F)
evaluate(opconjugate, iiconjugate)$BasicOperatorFunctions1(F)
-- evaluate(opGamma2 , iiGamma2 )$BasicOperatorFunctions1(F)
evaluate(opBeta , iiBeta )$BasicOperatorFunctions1(F)
evaluate(opdigamma , iidigamma )$BasicOperatorFunctions1(F)
evaluate(oppolygamma , iipolygamma)$BasicOperatorFunctions1(F)
evaluate(opBesselJ , iiBesselJ )$BasicOperatorFunctions1(F)
evaluate(opBesselY , iiBesselY )$BasicOperatorFunctions1(F)
evaluate(opBesselI , iiBesselI )$BasicOperatorFunctions1(F)
evaluate(opBesselK , iiBesselK )$BasicOperatorFunctions1(F)
evaluate(opAiryAi , iiAiryAi )$BasicOperatorFunctions1(F)
evaluate(opAiryAiPrime, iiAiryAiPrime)$BasicOperatorFunctions1(F)
evaluate(opAiryBi , iiAiryBi )$BasicOperatorFunctions1(F)
evaluate(opAiryBiPrime, iiAiryBiPrime)$BasicOperatorFunctions1(F)
evaluate(opLambertW, iLambertW)$BasicOperatorFunctions1(F)
evaluate(opPolylog, iPolylog)$BasicOperatorFunctions1(F)
evaluate(opWeierstrassP, iiWeierstrassP)$BasicOperatorFunctions1(F)
evaluate(opWeierstrassPPrime,
iiWeierstrassPPrime)$BasicOperatorFunctions1(F)
evaluate(opWeierstrassSigma, iiWeierstrassSigma)$BasicOperatorFunctions1(F)
evaluate(opWeierstrassZeta, iiWeierstrassZeta)$BasicOperatorFunctions1(F)
evaluate(opHypergeometricF, iiHypergeometricF)$BasicOperatorFunctions1(F)
evaluate(opMeijerG, iiMeijerG)$BasicOperatorFunctions1(F)
diff1(op : OP, n : F, x : F) : F ==
dm := dummy
kernel(opdiff, [op [dm, x], dm, n])
iBesselJ(l : List F, t : SE) : F ==
n := first l; x := second l
differentiate(n, t)diff1(opBesselJ, n, x)
+ differentiate(x, t) ahalf * (besselJ (n-1, x) - besselJ (n+1, x))
iBesselY(l : List F, t : SE) : F ==
n := first l; x := second l
differentiate(n, t)diff1(opBesselY, n, x)
+ differentiate(x, t) ahalf * (besselY (n-1, x) - besselY (n+1, x))
iBesselI(l : List F, t : SE) : F ==
n := first l; x := second l
differentiate(n, t)diff1(opBesselI, n, x)
+ differentiate(x, t) ahalf * (besselI (n-1, x) + besselI (n+1, x))
iBesselK(l : List F, t : SE) : F ==
n := first l; x := second l
differentiate(n, t)diff1(opBesselK, n, x)
- differentiate(x, t) ahalf * (besselK (n-1, x) + besselK (n+1, x))
dPolylog(l : List F, t : SE) : F ==
s := first l; x := second l
differentiate(s, t)diff1(opPolylog, s, x)
+ differentiate(x, t)polylog(s-1, x)/x
ipolygamma(l : List F, x : SE) : F ==
import from List(Symbol)
member?(x, variables first l) =>
error "cannot differentiate polygamma with respect to the first argument"
n := first l; y := second l
differentiate(y, x)polygamma(n+1, y)
iBetaGrad1(l : List F) : F ==
x := first l; y := second l
Beta(x, y)(digamma x - digamma(x+y))
iBetaGrad2(l : List F) : F ==
x := first l; y := second l
Beta(x, y)*(digamma y - digamma(x+y))
if F has ElementaryFunctionCategory then
iGamma2(l : List F, t : SE) : F ==
a := first l; x := second l
differentiate(a, t)diff1(opGamma2, a, x)
- differentiate(x, t) x ^ (a - 1) * exp(-x)
setProperty(opGamma2, SPECIALDIFF, iGamma2@((List F, SE)->F)
pretend None)
inGamma2(li : List INP) : INP ==
convert cons(convert('Gamma), li)
input(opGamma2, inGamma2@((List INP) -> INP))
dLambertW(x : F) : F ==
lw := lambertW(x)
lw/(x*(1+lw))
iWeierstrassPGrad1(l : List F) : F ==
g2 := first l
g3 := second l
x := third l
delta := g2^3 - 27g3^2
wp := weierstrassP(g2, g3, x)
(weierstrassPPrime(g2, g3, x)(-9ahalfg3
weierstrassZeta(g2, g3, x) + afourthg2^2x)
- 9g3wp^2 + ahalfg2^2wp + 3ahalfg2g3)/delta
iWeierstrassPGrad2(l : List F) : F ==
g2 := first l
g3 := second l
x := third l
delta := g2^3 - 27g3^2
wp := weierstrassP(g2, g3, x)
(weierstrassPPrime(g2, g3, x)(3g2weierstrassZeta(g2, g3, x)
- 9ahalfg3x) + 6g2wp^2 - 9g3*wp-g2^2)/delta
iWeierstrassPGrad3(l : List F) : F ==
weierstrassPPrime(first l, second l, third l)
iWeierstrassPPrimeGrad1(l : List F) : F ==
g2 := first l
g3 := second l
x := third l
delta := g2^3 - 27g3^2
wp := weierstrassP(g2, g3, x)
wpp := weierstrassPPrime(g2, g3, x)
wpp2 := 6wp^2 - ahalfg2
(wpp2(-9ahalfg3weierstrassZeta(g2, g3, x) + afourthg2^2x)
+ wpp(9ahalfg3wp + afourthg2^2) - 18g3wpwpp
+ ahalfg2^2*wpp)/delta
iWeierstrassPPrimeGrad2(l : List F) : F ==
g2 := first l
g3 := second l
x := third l
delta := g2^3 - 27g3^2
wp := weierstrassP(g2, g3, x)
wpp := weierstrassPPrime(g2, g3, x)
wpp2 := 6wp^2 - ahalfg2
(wpp2(3g2weierstrassZeta(g2, g3, x) - 9ahalfg3x)
+ wpp(-3g2wp - 9ahalfg3) + 12g2wpwpp - 9g3*wpp)/delta
iWeierstrassPPrimeGrad3(l : List F) : F ==
g2 := first l
6weierstrassP(g2, second l, third l)^2 - ahalfg2
iWeierstrassSigmaGrad1(l : List F) : F ==
g2 := first l
g3 := second l
x := third l
delta := g2^3 - 27g3^2
ws := weierstrassSigma(g2, g3, x)
wz := weierstrassZeta(g2, g3, x)
wsp := wzws
wsp2 := - weierstrassP(g2, g3, x)ws + wz^2ws
afourth(-9g3wsp2 - g2^2ws
- 3afourthg2g3x^2ws + g2^2x*wsp)/delta
iWeierstrassSigmaGrad2(l : List F) : F ==
g2 := first l
g3 := second l
x := third l
delta := g2^3 - 27g3^2
ws := weierstrassSigma(g2, g3, x)
wz := weierstrassZeta(g2, g3, x)
wsp := wzws
wsp2 := - weierstrassP(g2, g3, x)ws + wz^2ws
ahalf(3g2wsp2 + 9g3ws
+ afourthg2^2x^2ws - 9g3x*wsp)/delta
iWeierstrassSigmaGrad3(l : List F) : F ==
g2 := first l
g3 := second l
x := third l
weierstrassZeta(g2, g3, x)*weierstrassSigma(g2, g3, x)
iWeierstrassZetaGrad1(l : List F) : F ==
g2 := first l
g3 := second l
x := third l
delta := g2^3 - 27g3^2
wp := weierstrassP(g2, g3, x)
(ahalfweierstrassZeta(g2, g3, x)(9g3wp + ahalfg2^2)
- ahalfg2x(ahalfg2wp+3afourthg3)
+ 9afourthg3weierstrassPPrime(g2, g3, x))/delta
iWeierstrassZetaGrad2(l : List F) : F ==
g2 := first l
g3 := second l
x := third l
delta := g2^3 - 27g3^2
wp := weierstrassP(g2, g3, x)
(-3weierstrassZeta(g2, g3, x)(g2wp + 3ahalfg3) +
ahalfx(9g3wp + ahalfg2^2)
- 3ahalfg2weierstrassPPrime(g2, g3, x))/delta
iWeierstrassZetaGrad3(l : List F) : F ==
-weierstrassP(first l, second l, third l)
OF ==> OutputForm
SEX ==> SExpression
NNI ==> NonNegativeInteger
dconjugate(lo : List OF) : OF == overbar lo.1
display(opconjugate,dconjugate)
if F has RetractableTo(Integer) then
get_int_listf : List F -> List Integer
get_int_listo : (Integer, List OF) -> List Integer
get_int_listi : (Integer, List INP) -> List Integer
get_int_listf(lf : List F) : List Integer ==
map(z +-> retract(z)@Integer, lf)$ListFunctions2(F, Integer)
replace_i(lp : List F, v : F, i : NNI) : List F ==
concat(first(lp, (i - 1)::NNI), cons(v, rest(lp, i)))
iiHypergeometricF(l) ==
n := #l
z := l(n-2)
if z = 0 then
nn := (n - 2)::NNI
pq := rest(l, nn)
pqi := get_int_listf(pq)
p := first(pqi)
q := first(rest(pqi))
p <= q + 1 => return 1
kernel(opHypergeometricF, l)
idvsum(op : BasicOperator, n : Integer, l : List F, x : Symbol) : F ==
res : F := 0
for i in 1..n for a in l repeat
dm := dummy
nl := replace_i(l, dm, i)
res := res + differentiate(a, x)*kernel(opdiff, [op nl, dm, a])
res
dvhypergeom(l : List F, x : Symbol) : F ==
n := #l
nn := (n - 2)::NNI
pq := rest(l, nn)
pqi := get_int_listf(pq)
ol := l
l := first(l, nn)
l1 := reverse(l)
z := first(l1)
p := first(pqi)
q := first(rest(pqi))
aprod := 1@F
nl := []@(List F)
for i in 1..p repeat
a := first(l)
nl := cons(a + 1, nl)
aprod := aprod a
l := rest(l)
bprod := 1@F
for i in 1..q repeat
b := first(l)
nl := cons(b + 1, nl)
bprod := bprod b
l := rest(l)
nl0 := reverse!(nl)
nl1 := cons(z, pq)
nl := concat(nl0, nl1)
aprod := aprod/bprod
idvsum(opHypergeometricF, nn - 1, ol, x) +
differentiate(z, x)aprodopHypergeometricF(nl)
add_pairs_to_list(lp : List List F, l : List F) : List F ==
for p in lp repeat
#p ~= 2 => error "not a list of pairs"
l := cons(p(2), cons(p(1), l))
l
dvmeijer(l : List F, x : Symbol) : F ==
n := #l
nn := (n - 4)::NNI
l0 := l
nl := rest(l, nn)
nli := get_int_listf(nl)
l := first(l, nn)
l1 := reverse(l)
z := first(l1)
n1 := first(nli)
n2 := nli(2)
a := first l
sign : F := 1
if n1 > 0 or n2 > 0 then
na := a - 1
if n1 = 0 then sign := -1
l2 := cons(na, rest l)
else
na := a
if nli(3) > 0 then sign := -1
l2 := cons(a + 1, rest l)
nm : F := opMeijerG(concat(l2, nl))
om : F := opMeijerG(l0)
idvsum(opMeijerG, nn - 1, l0, x) +
differentiate(z, x)(signnm + na*om)/z
get_if_list(n : Integer, lf : List INP) : List List INP ==
a := []@(List INP)
for i in 1..n repeat
a := cons(first(lf), a)
lf := rest(lf)
a := cons(convert('construct), reverse!(a))
[a, lf]
get_if_lists(ln : List Integer, lf : List INP) : List List INP ==
rl := []@(List List INP)
for n in ln repeat
al := get_if_list(n, lf)
rl := cons(first(al), rl)
lf := first(rest(al))
rl := reverse!(rl)
cons(lf, rl)
get_int_listi(n : Integer, lo : List INP) : List Integer ==
n0 := (#lo - n)::NNI
lo := rest(lo, n0)
rl := []@(List Integer)
for i in 1..n repeat
p := integer(first(lo) pretend SEX)$SEX
rl := cons(p, rl)
lo := rest(lo)
rl := reverse!(rl)
rl
get_of_list(n : Integer, lo : List OF) : List List OF ==
a := []@(List OF)
for i in 1..n repeat
a := cons(first(lo), a)
lo := rest(lo)
a := reverse!(a)
[a, lo]
get_of_lists(ln : List Integer, lo : List OF) : List List OF ==
rl := []@(List List OF)
for n in ln repeat
al := get_of_list(n, lo)
rl := cons(first(al), rl)
lo := first(rest(al))
rl := reverse!(rl)
cons(lo, rl)
get_int_listo(n : Integer, lo : List OF) : List Integer ==
n0 := (#lo - n)::NNI
lo := rest(lo, n0)
rl := []@(List Integer)
for i in 1..n repeat
p := integer(first(lo) pretend SEX)$SEX
rl := cons(p, rl)
lo := rest(lo)
rl := reverse!(rl)
rl
dhyper0(op : OF, lo : List OF) : OF ==
n0 := (#lo - 2)::NNI
pql := get_int_listo(2, lo)
lo := first(lo, n0)
al := get_of_lists(pql, lo)
lo := first(al)
al := rest(al)
a := first al
b := first(rest(al))
z := first(lo)
prefix(op, [bracket a, bracket b, z])
dhyper(lo : List OF) : OF ==
dhyper0(hypergeometricF, lo)
ddhyper(lo : List OF) : OF ==
dhyper0(first lo, rest lo)
dmeijer0(op : OF, lo : List OF) : OF ==
n0 := (#lo - 4)::NNI
nl := get_int_listo(4, lo)
lo := first(lo, n0)
al := get_of_lists(nl, lo)
lo := first(al)
al := rest(al)
z := first(lo)
prefix(op, concat(
map(bracket, al)$ListFunctions2(List OF, OF), [z]))
dmeijer(lo : List OF) : OF ==
dmeijer0('meijerG::OF, lo)
ddmeijer(lo : List OF) : OF ==
dmeijer0(first lo, rest lo)
setProperty(opHypergeometricF, '%diffDisp,
ddhyper@(List OF -> OF) pretend None)
setProperty(opMeijerG, '%diffDisp,
ddmeijer@(List OF -> OF) pretend None)
display(opHypergeometricF, dhyper)
display(opMeijerG, dmeijer)
setProperty(opHypergeometricF, SPECIALDIFF,
dvhypergeom@((List F, Symbol)->F) pretend None)
setProperty(opMeijerG, SPECIALDIFF, dvmeijer@((List F, Symbol)->F)
pretend None)
inhyper(lf : List INP) : INP ==
pqi := get_int_listi(2, lf)
al := get_if_lists(pqi, lf)
lf := first(al)
al := rest(al)
a := first al
ai : INP := convert(a)
b := first(rest(al))
bi : INP := convert(b)
zi := first(lf)
li : List INP := [convert('hypergeometricF), ai, bi, zi]
convert(li)
input(opHypergeometricF, inhyper@((List INP) -> INP))
inmeijer(lf : List INP) : INP ==
pqi := get_int_listi(4, lf)
al := get_if_lists(pqi, lf)
lf := first(al)
al := rest(al)
a := first al
ai : INP := convert(a)
al := rest(al)
b := first(al)
bi : INP := convert(b)
al := rest(al)
c := first(al)
ci : INP := convert(c)
al := rest(al)
d := first(al)
di : INP := convert(d)
zi := first(lf)
li : List INP := [convert('meijerG), ai, bi, ci, di, zi]
convert(li)
input(opMeijerG, inmeijer@((List INP) -> INP))
else
iiHypergeometricF(l) == kernel(opHypergeometricF, l)
iiMeijerG(l) == kernel(opMeijerG, l)
d_eis(x : F) : F == -kernel(op_eis, x) + 1/x
if F has TranscendentalFunctionCategory
and F has RadicalCategory then
d_erfs(x : F) : F == 2xkernel(op_erfs, x) - 2::F/sqrt(pi())
d_erfis(x : F) : F == -2xkernel(op_erfis, x) + 2::F/sqrt(pi())
derivative(op_erfs, d_erfs)
derivative(op_erfis, d_erfis)
derivative(opabs, (x : F) : F +-> conjugate(x)inv(2abs(x)))
derivative(opconjugate, (x : F) : F +-> 0)
derivative(opGamma, (x : F) : F +-> digamma(x)*Gamma(x))
derivative(op_log_gamma, (x : F) : F +-> digamma(x))
derivative(opBeta, [iBetaGrad1, iBetaGrad2])
derivative(opdigamma, (x : F) : F +-> polygamma(1, x))
derivative(op_eis, d_eis)
derivative(opAiryAi, (x : F) : F +-> airyAiPrime(x))
derivative(opAiryAiPrime, (x : F) : F +-> xairyAi(x))
derivative(opAiryBi, (x : F) : F +-> airyBiPrime(x))
derivative(opAiryBiPrime, (x : F) : F +-> xairyBi(x))
derivative(opLambertW, dLambertW)
derivative(opWeierstrassP, [iWeierstrassPGrad1, iWeierstrassPGrad2,
iWeierstrassPGrad3])
derivative(opWeierstrassPPrime, [iWeierstrassPPrimeGrad1,
iWeierstrassPPrimeGrad2, iWeierstrassPPrimeGrad3])
derivative(opWeierstrassSigma, [iWeierstrassSigmaGrad1,
iWeierstrassSigmaGrad2, iWeierstrassSigmaGrad3])
derivative(opWeierstrassZeta, [iWeierstrassZetaGrad1,
iWeierstrassZetaGrad2, iWeierstrassZetaGrad3])
setProperty(oppolygamma, SPECIALDIFF, ipolygamma@((List F, SE)->F)
pretend None)
setProperty(opBesselJ, SPECIALDIFF, iBesselJ@((List F, SE)->F)
pretend None)
setProperty(opBesselY, SPECIALDIFF, iBesselY@((List F, SE)->F)
pretend None)
setProperty(opBesselI, SPECIALDIFF, iBesselI@((List F, SE)->F)
pretend None)
setProperty(opBesselK, SPECIALDIFF, iBesselK@((List F, SE)->F)
pretend None)
setProperty(opPolylog, SPECIALDIFF, dPolylog@((List F, SE)->F)
pretend None)
\end{spad}
Testing Display
\begin{axiom}
conjugate(_\sigma[_\alpha])
)set output algebra on
conjugate(x)
conjugate(a+%i*b)
)set output algebra off
\end{axiom}
Assertions
\begin{axiom}
test(conjugate abs x = abs x)
test(abs conjugate x = abs x)
test(conjugate conjugate x = x)
test(conjugate(a+b)=conjugate(a)+conjugate(b))
test(conjugate(ab)=conjugate(a)conjugate(b))
test(abs(2::Expression Integer)=2)
test(abs(-2::Expression Integer)=2)
test(abs(3.1415::Expression Float)=3.1415)
test(abs(1$Expression Complex Integer)=1)
test(abs((1+%i)::Expression Complex Float)=abs(1+%i))
c:=(1+%i)::Expression Complex Integer
test(abs(c)=sqrt(cconjugate(c)))
test(differentiate(conjugate(x),x)=0)
differentiate(abs(x),x)=differentiate(sqrt(xconjugate(x)),x)
test(differentiate(abs(x)^2,x)=conjugate(x))
\end{axiom}
Roots of Unity
\begin{axiom}
test(conjugate(sqrt(-1))::AlgebraicNumber = -sqrt(-1))
test(conjugate(sqrt(-1))+sqrt(-1)=0)
conjugate complexNumeric nthRoot(-1,3)
complexNumeric conjugate nthRoot(-1,3)
conjugate sqrt(complex(-1,0))
\end{axiom}
General roots
\begin{axiom}
norm(conjugate complexNumeric sqrt(-1+sqrt(-1))-complexNumeric conjugate sqrt(-1+sqrt(-1)))<10.0^(-precision()/2)
norm(conjugate complexNumeric sqrt(-1+%i)-complexNumeric conjugate sqrt(-1+%i))<10.0^(-precision()/2)
\end{axiom}
\begin{axiom}
)clear completely
)lib FSPECX
\end{axiom}
On 8 September 2014 15:15, Waldek Hebisch wrote:
Definitions you gave are debatable. In particular
D(conjugate(x), x) = 0 may lead to troubles. For example,
D(conjugate(conjugate(x)), x) should be equal to D(x, x) = 1
regardless how we compute it. Consider:
\begin{axiom}
D(conjugate(conjugate(x)), x)
y := operator 'y
conjugate(y(x))
D(conjugate(y(x)), x)
\end{axiom}
substituting conjugate(x) for y(x) and derivative of
conjugate(x) for derivative of y(x) should give derivative
of conjugate(conjugate(x)).
If what I wrote looks like nitpicking let me note that FriCAS
blindly applies rather complex transformations. For example
during integration internal form is quite different than
user input and final result. Inconsistency in derivative
rule will bring all kinds of nasty bugs.
We probably can leave derivative of conjugate unevaluated.
But even that needs some thought to make sure there are
no inconsistency. Signaling error would be safe from
correctness point of view, but would significantly limit
usefulness - code handling expressions assumes that it
can freely compute derivatives, so when dealing with
conjugate we would routinely get errors deep inside
library code.
On On 11 September 2014 07:28, Waldek Hebisch wrote:
You clearly allow non-holomorphic arguments to conjugate, otherwise
conjugate(conjugate(x)) would be illegal. Rather, you assume
that conjugate will be always pushed onto variables/parameters
and that in context of differentiation we will substitute
only holomorphic functions for variables. But in other
context you allow non-holomorphic things. Pushing conjugate
to variables in itself is debatable:
\begin{axiom}
conjugate(log(-1))
eval(conjugate(log(x)), x = -1)
\end{axiom}
you get different result depending on when exactly we plug in
constant argument to logarithm.
You have:
\begin{axiom}
conjugate(log(conjugate(x) + x))
\end{axiom}
but when real part of x is negative we are on the conventional
branch cut of logarithm and some folks may be upset by such
simplification (not that unlike previous example where problem
set was of lower dimension here we have problem on open set).
The reason that I wanted D(conjugate(x), x) = 0 is to have D
correspond to the first Wirtinger derivative that I mentioned in
another email chain. It is not clear to me that this could result in
nasty bugs. Maybe this is possible in the cases where the chain rule
is applied since in that case the conjugate Wirtinger derivative would
be required.
But chain rule is applied automatically when computing derivatives.
Consider:
\begin{axiom}
D(abs(x + conjugate(x)), x)
\end{axiom}
complex (Wirtinger) derivative of the above is twice of the above.
Actually, it seems that even leaving derivatives of conjugate
unevaluated we need to be careful. Namely, given
a differential Ring R_0 and any formal operation f we can form
differential ring R_1 where f and all its derivative remain
unevaluated. But we want to have some simplification
so we divide R_1 by appropriate equivalence relation.
For the result to be a differential ring equivalence
classes should be cosets of a differential ideal, in
particular set of elements of R_1 equivalent to 0
should be a differential ideal. So we need to make
sure that simplifications are consistent with
derivative.
\begin{axiom}
)clear completely
)lib FSPECX
\end{axiom}
I would like to collect some examples of such errors.
With your current code:
\begin{axiom}
normalize(exp(conjugate(x)+x)+exp(x) + exp(conjugate(x)))
\end{axiom}
With derivative of conjugate changes to error:
\begin{axiom}
integrate(exp(conjugate(y)*x), x)
\end{axiom}
\begin{axiom}
normalize(exp(conjugate(y)*x)+exp(x))
\end{axiom}
\begin{axiom}
limit(exp(conjugate(y)*x), x=%plusInfinity)
\end{axiom}
\begin{axiom}
series(conjugate(x), x=1)
\end{axiom}
\begin{axiom}
)lib FSPECX
\end{axiom}
\begin{axiom}
conjugate(log(conjugate(x) + x))
x:=(1+%i)
x+conjugate(x)
conjugate(log(conjugate(x) + x))
x:=(-1+%i)
x+conjugate(x)
conjugate(log(conjugate(x) + x))
\end{axiom}
\begin{axiom}
)lib FSPECX
\end{axiom}
\begin{axiom}
conjugate(log(complex(-1.0,0)))
log(conjugate(complex(-1.0,0)))
\end{axiom}
Some or all expressions may not have rendered properly,
because Axiom returned the following error:
Error: export HOME=/var/zope2/var/LatexWiki; ulimit -t 600; export LD_LIBRARY_PATH=/usr/local/lib/fricas/target/x86_64-linux-gnu/lib; LANG=en_US.UTF-8 /usr/local/lib/fricas/target/x86_64-linux-gnu/bin/fricas -nosman < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6353881970213761161-25px.axm
Heap exhausted during garbage collection: 0 bytes available, 16 requested.
Immobile Object Counts
Gen layout fdefn symbol code Boxed Cons Raw Code SmMix Mixed LgRaw LgCode LgMix Waste% Alloc Trig Dirty GCs Mem-age
3 0 0 0 0 0 20390 2 0 2 3 0 0 0 0.8 662853872 418541194 20397 1 1.4502
4 0 113 0 420 10 11421 22 1 5 10 0 0 0 0.8 372707744 2000000 9755 0 0.1511
5 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0 0 2000000 0 0 0.0000
6 750 25588 26806 25955 472 217 66 5 51 16 0 0 75 2.4 28859328 2000000 16 0 0.0000
Tot 750 25701 26806 26375 482 32028 90 6 58 29 0 0 75 0.9 1064420944 [99.1% of 1073741824 max]
GC control variables:
GC-INHIBIT = true
GC-PENDING = true
STOP-FOR-GC-PENDING = false
fatal error encountered in SBCL pid 698417 tid 698417:
Heap exhausted, game over.
Error opening /dev/tty: No such device or address
Checking for foreign routines
FRICAS="/usr/local/lib/fricas/target/x86_64-linux-gnu"
spad-lib="/usr/local/lib/fricas/target/x86_64-linux-gnu/lib/libspad.so"
foreign routines found
openServer result -2
FriCAS Computer Algebra System
Version: FriCAS 1.3.10 built with sbcl 2.2.9.debian
Timestamp: Wed 10 Jan 02:19:45 CET 2024
-----------------------------------------------------------------------------
Issue )copyright to view copyright notices.
Issue )summary for a summary of useful system commands.
Issue )quit to leave FriCAS and return to shell.
-----------------------------------------------------------------------------
(1) -> (1) -> (1) -> (1) -> (1) -> (1) -> <spad>
)abbrev package FSPECX FunctionalSpecialFunction
)boot $tryRecompileArguments := nil
++ Provides the special functions
++ Author: Manuel Bronstein
++ Date Created: 18 Apr 1989
++ Date Last Updated: 4 October 1993
++ Description: Provides some special functions over an integral domain.
++ Keywords: special, function.
FunctionalSpecialFunction(R, F) : Exports == Implementation where
R : Join(Comparable, IntegralDomain)
F : FunctionSpace R
OP ==> BasicOperator
K ==> Kernel F
SE ==> Symbol
SPECIALDIFF ==> '%specialDiff
Exports ==> with
belong? : OP -> Boolean
++ belong?(op) is true if op is a special function operator;
operator : OP -> OP
++ operator(op) returns a copy of op with the domain-dependent
++ properties appropriate for F;
++ error if op is not a special function operator
abs : F -> F
++ abs(f) returns the absolute value operator applied to f
conjugate: F -> F
++ conjugate(f) returns the conjugate value operator applied to f
Gamma : F -> F
++ Gamma(f) returns the formal Gamma function applied to f
Gamma : (F, F) -> F
++ Gamma(a, x) returns the incomplete Gamma function applied to a and x
Beta : (F, F) -> F
++ Beta(x, y) returns the beta function applied to x and y
digamma : F->F
++ digamma(x) returns the digamma function applied to x
polygamma : (F, F) ->F
++ polygamma(x, y) returns the polygamma function applied to x and y
besselJ : (F, F) -> F
++ besselJ(x, y) returns the besselj function applied to x and y
besselY : (F, F) -> F
++ besselY(x, y) returns the bessely function applied to x and y
besselI : (F, F) -> F
++ besselI(x, y) returns the besseli function applied to x and y
besselK : (F, F) -> F
++ besselK(x, y) returns the besselk function applied to x and y
airyAi : F -> F
++ airyAi(x) returns the Airy Ai function applied to x
airyAiPrime : F -> F
++ airyAiPrime(x) returns the derivative of Airy Ai function applied to x
airyBi : F -> F
++ airyBi(x) returns the Airy Bi function applied to x
airyBiPrime : F -> F
++ airyBiPrime(x) returns the derivative of Airy Bi function applied to x
lambertW : F -> F
++ lambertW(x) is the Lambert W function at x
polylog : (F, F) -> F
++ polylog(s, x) is the polylogarithm of order s at x
weierstrassP : (F, F, F) -> F
++ weierstrassP(g2, g3, x)
weierstrassPPrime : (F, F, F) -> F
++ weierstrassPPrime(g2, g3, x)
weierstrassSigma : (F, F, F) -> F
++ weierstrassSigma(g2, g3, x)
weierstrassZeta : (F, F, F) -> F
++ weierstrassZeta(g2, g3, x)
-- weierstrassPInverse : (F, F, F) -> F
-- ++ weierstrassPInverse(g2, g3, z) is the inverse of Weierstrass
-- ++ P function, defined by the formula
-- ++ \spad{weierstrassP(g2, g3, weierstrassPInverse(g2, g3, z)) = z}
whittakerM : (F, F, F) -> F
++ whittakerM(k, m, z) is the Whittaker M function
whittakerW : (F, F, F) -> F
++ whittakerW(k, m, z) is the Whittaker W function
angerJ : (F, F) -> F
++ angerJ(v, z) is the Anger J function
weberE : (F, F) -> F
++ weberE(v, z) is the Weber E function
struveH : (F, F) -> F
++ struveH(v, z) is the Struve H function
struveL : (F, F) -> F
++ struveL(v, z) is the Struve L function defined by the formula
++ \spad{struveL(v, z) = -%i^exp(-v%pi%i/2)struveH(v, %iz)}
hankelH1 : (F, F) -> F
++ hankelH1(v, z) is first Hankel function (Bessel function of
++ the third kind)
hankelH2 : (F, F) -> F
++ hankelH2(v, z) is the second Hankel function (Bessel function of
++ the third kind)
lommelS1 : (F, F, F) -> F
++ lommelS1(mu, nu, z) is the Lommel s function
lommelS2 : (F, F, F) -> F
++ lommelS2(mu, nu, z) is the Lommel S function
kummerM : (F, F, F) -> F
++ kummerM(a, b, z) is the Kummer M function
kummerU : (F, F, F) -> F
++ kummerU(a, b, z) is the Kummer U function
legendreP : (F, F, F) -> F
++ legendreP(nu, mu, z) is the Legendre P function
legendreQ : (F, F, F) -> F
++ legendreQ(nu, mu, z) is the Legendre Q function
kelvinBei : (F, F) -> F
++ kelvinBei(v, z) is the Kelvin bei function defined by equality
++ \spad{kelvinBei(v, z) = imag(besselJ(v, exp(3%pi%i/4)z))}
++ for z and v real
kelvinBer : (F, F) -> F
++ kelvinBer(v, z) is the Kelvin ber function defined by equality
++ \spad{kelvinBer(v, z) = real(besselJ(v, exp(3%pi%i/4)z))}
++ for z and v real
kelvinKei : (F, F) -> F
++ kelvinKei(v, z) is the Kelvin kei function defined by equality
++ \spad{kelvinKei(v, z) =
++ imag(exp(-v%pi%i/2)besselK(v, exp(%pi%i/4)z))}
++ for z and v real
kelvinKer : (F, F) -> F
++ kelvinKer(v, z) is the Kelvin kei function defined by equality
++ \spad{kelvinKer(v, z) =
++ real(exp(-v%pi%i/2)besselK(v, exp(%pi%i/4)z))}
++ for z and v real
ellipticK : F -> F
++ ellipticK(m) is the complete elliptic integral of the
++ first kind: \spad{ellipticK(m) =
++ integrate(1/sqrt((1-t^2)(1-mt^2)), t = 0..1)}
ellipticE : F -> F
++ ellipticE(m) is the complete elliptic integral of the
++ second kind: \spad{ellipticE(m) =
++ integrate(sqrt(1-mt^2)/sqrt(1-t^2), t = 0..1)}
ellipticE : (F, F) -> F
++ ellipticE(z, m) is the incomplete elliptic integral of the
++ second kind: \spad{ellipticE(z, m) =
++ integrate(sqrt(1-mt^2)/sqrt(1-t^2), t = 0..z)}
ellipticF : (F, F) -> F
++ ellipticF(z, m) is the incomplete elliptic integral of the
++ first kind : \spad{ellipticF(z, m) =
++ integrate(1/sqrt((1-t^2)(1-mt^2)), t = 0..z)}
ellipticPi : (F, F, F) -> F
++ ellipticPi(z, n, m) is the incomplete elliptic integral of
++ the third kind: \spad{ellipticPi(z, n, m) =
++ integrate(1/((1-nt^2)sqrt((1-t^2)(1-mt^2))), t = 0..z)}
jacobiSn : (F, F) -> F
++ jacobiSn(z, m) is the Jacobi elliptic sn function, defined
++ by the formula \spad{jacobiSn(ellipticF(z, m), m) = z}
jacobiCn : (F, F) -> F
++ jacobiCn(z, m) is the Jacobi elliptic cn function, defined
++ by \spad{jacobiCn(z, m)^2 + jacobiSn(z, m)^2 = 1} and
++ \spad{jacobiCn(0, m) = 1}
jacobiDn : (F, F) -> F
++ jacobiDn(z, m) is the Jacobi elliptic dn function, defined
++ by \spad{jacobiDn(z, m)^2 + m*jacobiSn(z, m)^2 = 1} and
++ \spad{jacobiDn(0, m) = 1}
jacobiZeta : (F, F) -> F
++ jacobiZeta(z, m) is the Jacobi elliptic zeta function, defined
++ by \spad{D(jacobiZeta(z, m), z) =
++ jacobiDn(z, m)^2 - ellipticE(m)/ellipticK(m)} and
++ \spad{jacobiZeta(0, m) = 0}.
jacobiTheta : (F, F) -> F
++ jacobiTheta(q, z) is the third Jacobi Theta function
lerchPhi : (F, F, F) -> F
++ lerchPhi(z, s, a) is the Lerch Phi function
riemannZeta : F -> F
++ riemannZeta(z) is the Riemann Zeta function
charlierC : (F, F, F) -> F
++ charlierC(n, a, z) is the Charlier polynomial
hermiteH : (F, F) -> F
++ hermiteH(n, z) is the Hermite polynomial
jacobiP : (F, F, F, F) -> F
++ jacobiP(n, a, b, z) is the Jacobi polynomial
laguerreL: (F, F, F) -> F
++ laguerreL(n, a, z) is the Laguerre polynomial
meixnerM : (F, F, F, F) -> F
++ meixnerM(n, b, c, z) is the Meixner polynomial
if F has RetractableTo(Integer) then
hypergeometricF : (List F, List F, F) -> F
++ hypergeometricF(la, lb, z) is the generalized hypergeometric
++ function
meijerG : (List F, List F, List F, List F, F) -> F
++ meijerG(la, lb, lc, ld, z) is the meijerG function
-- Functions below should be local but conditional
iiGamma : F -> F
++ iiGamma(x) should be local but conditional;
iiabs : F -> F
++ iiabs(x) should be local but conditional;
iiconjugate: F -> F
++ iiconjugate(x) should be local but conditional;
iiBeta : List F -> F
++ iiBeta(x) should be local but conditional;
iidigamma : F -> F
++ iidigamma(x) should be local but conditional;
iipolygamma : List F -> F
++ iipolygamma(x) should be local but conditional;
iiBesselJ : List F -> F
++ iiBesselJ(x) should be local but conditional;
iiBesselY : List F -> F
++ iiBesselY(x) should be local but conditional;
iiBesselI : List F -> F
++ iiBesselI(x) should be local but conditional;
iiBesselK : List F -> F
++ iiBesselK(x) should be local but conditional;
iiAiryAi : F -> F
++ iiAiryAi(x) should be local but conditional;
iiAiryAiPrime : F -> F
++ iiAiryAiPrime(x) should be local but conditional;
iiAiryBi : F -> F
++ iiAiryBi(x) should be local but conditional;
iiAiryBiPrime : F -> F
++ iiAiryBiPrime(x) should be local but conditional;
iAiryAi : F -> F
++ iAiryAi(x) should be local but conditional;
iAiryAiPrime : F -> F
++ iAiryAiPrime(x) should be local but conditional;
iAiryBi : F -> F
++ iAiryBi(x) should be local but conditional;
iAiryBiPrime : F -> F
++ iAiryBiPrime(x) should be local but conditional;
iiHypergeometricF : List F -> F
++ iiHypergeometricF(l) should be local but conditional;
iiPolylog : (F, F) -> F
++ iiPolylog(x, s) should be local but conditional;
iLambertW : F -> F
++ iLambertW(x) should be local but conditional;
Implementation ==> add
SPECIAL := 'special
INP ==> InputForm
SPECIALINPUT ==> '%specialInput
iabs : F -> F
iGamma : F -> F
iBeta : (F, F) -> F
idigamma : F -> F
iiipolygamma : (F, F) -> F
iiiBesselJ : (F, F) -> F
iiiBesselY : (F, F) -> F
iiiBesselI : (F, F) -> F
iiiBesselK : (F, F) -> F
iPolylog : List F -> F
iWeierstrassP : (F, F, F) -> F
iWeierstrassPPrime : (F, F, F) -> F
iWeierstrassSigma : (F, F, F) -> F
iWeierstrassZeta : (F, F, F) -> F
iiWeierstrassP : List F -> F
iiWeierstrassPPrime : List F -> F
iiWeierstrassSigma : List F -> F
iiWeierstrassZeta : List F -> F
iiMeijerG : List F -> F
opabs := operator('abs)$CommonOperators
opconjugate := operator('conjugate)$CommonOperators
opGamma := operator('Gamma)$CommonOperators
opGamma2 := operator('Gamma2)$CommonOperators
opBeta := operator('Beta)$CommonOperators
opdigamma := operator('digamma)$CommonOperators
oppolygamma := operator('polygamma)$CommonOperators
opBesselJ := operator('besselJ)$CommonOperators
opBesselY := operator('besselY)$CommonOperators
opBesselI := operator('besselI)$CommonOperators
opBesselK := operator('besselK)$CommonOperators
opAiryAi := operator('airyAi)$CommonOperators
opAiryAiPrime := operator('airyAiPrime)$CommonOperators
opAiryBi := operator('airyBi)$CommonOperators
opAiryBiPrime := operator('airyBiPrime)$CommonOperators
opLambertW := operator('lambertW)$CommonOperators
opPolylog := operator('polylog)$CommonOperators
opWeierstrassP := operator('weierstrassP)$CommonOperators
opWeierstrassPPrime := operator('weierstrassPPrime)$CommonOperators
opWeierstrassSigma := operator('weierstrassSigma)$CommonOperators
opWeierstrassZeta := operator('weierstrassZeta)$CommonOperators
opHypergeometricF := operator('hypergeometricF)$CommonOperators
opMeijerG := operator('meijerG)$CommonOperators
opCharlierC := operator('charlierC)$CommonOperators
opHermiteH := operator('hermiteH)$CommonOperators
opJacobiP := operator('jacobiP)$CommonOperators
opLaguerreL := operator('laguerreL)$CommonOperators
opMeixnerM := operator('meixnerM)$CommonOperators
op_log_gamma := operator('%logGamma)$CommonOperators
op_eis := operator('%eis)$CommonOperators
op_erfs := operator('%erfs)$CommonOperators
op_erfis := operator('%erfis)$CommonOperators
abs x == opabs x
conjugate x == opconjugate x
Gamma(x) == opGamma(x)
Gamma(a, x) == opGamma2(a, x)
Beta(x, y) == opBeta(x, y)
digamma x == opdigamma(x)
polygamma(k, x)== oppolygamma(k, x)
besselJ(a, x) == opBesselJ(a, x)
besselY(a, x) == opBesselY(a, x)
besselI(a, x) == opBesselI(a, x)
besselK(a, x) == opBesselK(a, x)
airyAi(x) == opAiryAi(x)
airyAiPrime(x) == opAiryAiPrime(x)
airyBi(x) == opAiryBi(x)
airyBiPrime(x) == opAiryBiPrime(x)
lambertW(x) == opLambertW(x)
polylog(s, x) == opPolylog(s, x)
weierstrassP(g2, g3, x) == opWeierstrassP(g2, g3, x)
weierstrassPPrime(g2, g3, x) == opWeierstrassPPrime(g2, g3, x)
weierstrassSigma(g2, g3, x) == opWeierstrassSigma(g2, g3, x)
weierstrassZeta(g2, g3, x) == opWeierstrassZeta(g2, g3, x)
if F has RetractableTo(Integer) then
hypergeometricF(a, b, z) ==
nai := #a
nbi := #b
z = 0 and nai <= nbi + 1 => 1
p := (#a)::F
q := (#b)::F
opHypergeometricF concat(concat(a, concat(b, [z])), [p, q])
meijerG(a, b, c, d, z) ==
n1 := (#a)::F
n2 := (#b)::F
m1 := (#c)::F
m2 := (#d)::F
opMeijerG concat(concat(a, concat(b,
concat(c, concat(d, [z])))), [n1, n2, m1, m2])
import from List Kernel(F)
opdiff := operator(operator('%diff)$CommonOperators)$F
dummy ==> new()$SE :: F
ahalf : F := recip(2::F)::F
athird : F := recip(3::F)::F
afourth : F := recip(4::F)::F
asixth : F := recip(6::F)::F
twothirds : F := 2athird
threehalfs : F := 3ahalf
-- Helpers for partially defined derivatives
grad2(l : List F, t : SE, op : OP, d2 : (F, F) -> F ) : F ==
x1 := l(1)
x2 := l(2)
dm := dummy
differentiate(x1, t)*kernel(opdiff, [op [dm, x2], dm, x1])
+ differentiate(x2, t)*d2(x1, x2)
grad3(l : List F, t : SE, op : OP, d3 : (F, F, F) -> F ) : F ==
x1 := l(1)
x2 := l(2)
x3 := l(3)
dm1 := dummy
dm2 := dummy
differentiate(x1, t)*kernel(opdiff, [op [dm1, x2, x3], dm1, x1])
+ differentiate(x2, t)*kernel(opdiff, [op [x1, dm2, x3], dm2, x2])
+ differentiate(x3, t)*d3(x1, x2, x3)
grad4(l : List F, t : SE, op : OP, d4 : (F, F, F, F) -> F ) : F ==
x1 := l(1)
x2 := l(2)
x3 := l(3)
x4 := l(4)
dm1 := dummy
dm2 := dummy
dm3 := dummy
kd1 := kernel(opdiff, [op [dm1, x2, x3, x4], dm1, x1])
kd2 := kernel(opdiff, [op [x1, dm2, x3, x4], dm2, x2])
kd3 := kernel(opdiff, [op [x1, x2, dm3, x4], dm3, x3])
differentiate(x1, t)kd1 + differentiate(x2, t)kd2 +
differentiate(x3, t)kd3 +
differentiate(x4, t)d4(x1, x2, x3, x4)
-- handle WeierstrassPInverse
)if false
opWeierstrassPInverse := operator('weierstrassPInverse)$CommonOperators
weierstrassPInverse(g2, g3, z) == opWeierstrassPInverse(g2, g3, z)
eWeierstrassPInverse(g2 : F, g3 : F, z : F) : F ==
kernel(opWeierstrassPInverse, [g2, g3, z])
elWeierstrassPInverse(l : List F) : F == eWeierstrassPInverse(l(1), l(2), l(3))
evaluate(opWeierstrassPInverse, elWeierstrassPInverse)$BasicOperatorFunctions1(F)
if F has RadicalCategory then
eWeierstrassPInverseGrad_g2(l : List F) : F ==
g2 := l(1)
g3 := l(2)
z := l(3)
error "unimplemented"
eWeierstrassPInverseGrad_g3(l : List F) : F ==
g2 := l(1)
g3 := l(2)
z := l(3)
error "unimplemented"
eWeierstrassPInverseGrad_z(l : List F) : F ==
g2 := l(1)
g3 := l(2)
z := l(3)
1/sqrt(4z^3 - g2z - g3)
derivative(opWeierstrassPInverse, [eWeierstrassPInverseGrad_g2,
eWeierstrassPInverseGrad_g3, eWeierstrassPInverseGrad_z])
)endif
-- handle WhittakerM
opWhittakerM := operator('whittakerM)$CommonOperators
whittakerM(k, m, z) == opWhittakerM(k, m, z)
eWhittakerM(k : F, m : F, z : F) : F ==
kernel(opWhittakerM, [k, m, z])
elWhittakerM(l : List F) : F == eWhittakerM(l(1), l(2), l(3))
evaluate(opWhittakerM, elWhittakerM)$BasicOperatorFunctions1(F)
eWhittakerMGrad_z(k : F, m : F, z : F) : F ==
(ahalf - k/z)whittakerM(k, m, z) +
(ahalf + k + m)whittakerM(k + 1, m, z)/z
dWhittakerM(l : List F, t : SE) : F ==
grad3(l, t, opWhittakerM, eWhittakerMGrad_z)
setProperty(opWhittakerM, SPECIALDIFF, dWhittakerM@((List F, SE)->F)
pretend None)
-- handle WhittakerW
opWhittakerW := operator('whittakerW)$CommonOperators
whittakerW(k, m, z) == opWhittakerW(k, m, z)
eWhittakerW(k : F, m : F, z : F) : F ==
kernel(opWhittakerW, [k, m, z])
elWhittakerW(l : List F) : F == eWhittakerW(l(1), l(2), l(3))
evaluate(opWhittakerW, elWhittakerW)$BasicOperatorFunctions1(F)
eWhittakerWGrad_z(k : F, m : F, z : F) : F ==
(ahalf - k/z)*whittakerW(k, m, z) - whittakerW(k + 1, m, z)/z
dWhittakerW(l : List F, t : SE) : F ==
grad3(l, t, opWhittakerW, eWhittakerWGrad_z)
setProperty(opWhittakerW, SPECIALDIFF, dWhittakerW@((List F, SE)->F)
pretend None)
-- handle AngerJ
opAngerJ := operator('angerJ)$CommonOperators
angerJ(v, z) == opAngerJ(v, z)
if F has TranscendentalFunctionCategory then
eAngerJ(v : F, z : F) : F ==
z = 0 => sin(vpi())/(vpi())
kernel(opAngerJ, [v, z])
elAngerJ(l : List F) : F == eAngerJ(l(1), l(2))
evaluate(opAngerJ, elAngerJ)$BasicOperatorFunctions1(F)
eAngerJGrad_z(v : F, z : F) : F ==
-angerJ(v + 1, z) + vangerJ(v, z)/z - sin(vpi())/(pi()*z)
dAngerJ(l : List F, t : SE) : F ==
grad2(l, t, opAngerJ, eAngerJGrad_z)
setProperty(opAngerJ, SPECIALDIFF, dAngerJ@((List F, SE)->F)
pretend None)
else
eeAngerJ(l : List F) : F == kernel(opAngerJ, l)
evaluate(opAngerJ, eeAngerJ)$BasicOperatorFunctions1(F)
-- handle WeberE
opWeberE := operator('weberE)$CommonOperators
weberE(v, z) == opWeberE(v, z)
if F has TranscendentalFunctionCategory then
eWeberE(v : F, z : F) : F ==
z = 0 => 2sin(ahalfvpi())^2/(vpi())
kernel(opWeberE, [v, z])
elWeberE(l : List F) : F == eWeberE(l(1), l(2))
evaluate(opWeberE, elWeberE)$BasicOperatorFunctions1(F)
eWeberEGrad_z(v : F, z : F) : F ==
-weberE(v + 1, z) + vweberE(v, z)/z - (1 - cos(vpi()))/(pi()*z)
dWeberE(l : List F, t : SE) : F ==
grad2(l, t, opWeberE, eWeberEGrad_z)
setProperty(opWeberE, SPECIALDIFF, dWeberE@((List F, SE)->F)
pretend None)
else
eeWeberE(l : List F) : F == kernel(opWeberE, l)
evaluate(opWeberE, eeWeberE)$BasicOperatorFunctions1(F)
-- handle StruveH
opStruveH := operator('struveH)$CommonOperators
struveH(v, z) == opStruveH(v, z)
eStruveH(v : F, z : F) : F ==
kernel(opStruveH, [v, z])
elStruveH(l : List F) : F == eStruveH(l(1), l(2))
evaluate(opStruveH, elStruveH)$BasicOperatorFunctions1(F)
if F has TranscendentalFunctionCategory
and F has RadicalCategory then
eStruveHGrad_z(v : F, z : F) : F ==
-struveH(v + 1, z) + vstruveH(v, z)/z +
(ahalfz)^v/(sqrt(pi())*Gamma(v + threehalfs))
dStruveH(l : List F, t : SE) : F ==
grad2(l, t, opStruveH, eStruveHGrad_z)
setProperty(opStruveH, SPECIALDIFF, dStruveH@((List F, SE)->F)
pretend None)
-- handle StruveL
opStruveL := operator('struveL)$CommonOperators
struveL(v, z) == opStruveL(v, z)
eStruveL(v : F, z : F) : F ==
kernel(opStruveL, [v, z])
elStruveL(l : List F) : F == eStruveL(l(1), l(2))
evaluate(opStruveL, elStruveL)$BasicOperatorFunctions1(F)
if F has TranscendentalFunctionCategory
and F has RadicalCategory then
eStruveLGrad_z(v : F, z : F) : F ==
struveL(v + 1, z) + vstruveL(v, z)/z +
(ahalfz)^v/(sqrt(pi())*Gamma(v + threehalfs))
dStruveL(l : List F, t : SE) : F ==
grad2(l, t, opStruveL, eStruveLGrad_z)
setProperty(opStruveL, SPECIALDIFF, dStruveL@((List F, SE)->F)
pretend None)
-- handle HankelH1
opHankelH1 := operator('hankelH1)$CommonOperators
hankelH1(v, z) == opHankelH1(v, z)
eHankelH1(v : F, z : F) : F ==
kernel(opHankelH1, [v, z])
elHankelH1(l : List F) : F == eHankelH1(l(1), l(2))
evaluate(opHankelH1, elHankelH1)$BasicOperatorFunctions1(F)
eHankelH1Grad_z(v : F, z : F) : F ==
-hankelH1(v + 1, z) + v*hankelH1(v, z)/z
dHankelH1(l : List F, t : SE) : F ==
grad2(l, t, opHankelH1, eHankelH1Grad_z)
setProperty(opHankelH1, SPECIALDIFF, dHankelH1@((List F, SE)->F)
pretend None)
-- handle HankelH2
opHankelH2 := operator('hankelH2)$CommonOperators
hankelH2(v, z) == opHankelH2(v, z)
eHankelH2(v : F, z : F) : F ==
kernel(opHankelH2, [v, z])
elHankelH2(l : List F) : F == eHankelH2(l(1), l(2))
evaluate(opHankelH2, elHankelH2)$BasicOperatorFunctions1(F)
eHankelH2Grad_z(v : F, z : F) : F ==
-hankelH2(v + 1, z) + v*hankelH2(v, z)/z
dHankelH2(l : List F, t : SE) : F ==
grad2(l, t, opHankelH2, eHankelH2Grad_z)
setProperty(opHankelH2, SPECIALDIFF, dHankelH2@((List F, SE)->F)
pretend None)
-- handle LommelS1
opLommelS1 := operator('lommelS1)$CommonOperators
lommelS1(m, v, z) == opLommelS1(m, v, z)
eLommelS1(m : F, v : F, z : F) : F ==
kernel(opLommelS1, [m, v, z])
elLommelS1(l : List F) : F == eLommelS1(l(1), l(2), l(3))
evaluate(opLommelS1, elLommelS1)$BasicOperatorFunctions1(F)
eLommelS1Grad_z(m : F, v : F, z : F) : F ==
-vlommelS1(m, v, z)/z + (m + v - 1)lommelS1(m - 1, v - 1, z)
dLommelS1(l : List F, t : SE) : F ==
grad3(l, t, opLommelS1, eLommelS1Grad_z)
setProperty(opLommelS1, SPECIALDIFF, dLommelS1@((List F, SE)->F)
pretend None)
-- handle LommelS2
opLommelS2 := operator('lommelS2)$CommonOperators
lommelS2(mu, nu, z) == opLommelS2(mu, nu, z)
eLommelS2(mu : F, nu : F, z : F) : F ==
kernel(opLommelS2, [mu, nu, z])
elLommelS2(l : List F) : F == eLommelS2(l(1), l(2), l(3))
evaluate(opLommelS2, elLommelS2)$BasicOperatorFunctions1(F)
eLommelS2Grad_z(m : F, v : F, z : F) : F ==
-vlommelS2(m, v, z)/z + (m + v - 1)lommelS2(m - 1, v - 1, z)
dLommelS2(l : List F, t : SE) : F ==
grad3(l, t, opLommelS2, eLommelS2Grad_z)
setProperty(opLommelS2, SPECIALDIFF, dLommelS2@((List F, SE)->F)
pretend None)
-- handle KummerM
opKummerM := operator('kummerM)$CommonOperators
kummerM(mu, nu, z) == opKummerM(mu, nu, z)
eKummerM(a : F, b : F, z : F) : F ==
z = 0 => 1
kernel(opKummerM, [a, b, z])
elKummerM(l : List F) : F == eKummerM(l(1), l(2), l(3))
evaluate(opKummerM, elKummerM)$BasicOperatorFunctions1(F)
eKummerMGrad_z(a : F, b : F, z : F) : F ==
((z + a - b)kummerM(a, b, z)+(b - a)kummerM(a - 1, b, z))/z
dKummerM(l : List F, t : SE) : F ==
grad3(l, t, opKummerM, eKummerMGrad_z)
setProperty(opKummerM, SPECIALDIFF, dKummerM@((List F, SE)->F)
pretend None)
-- handle KummerU
opKummerU := operator('kummerU)$CommonOperators
kummerU(a, b, z) == opKummerU(a, b, z)
eKummerU(a : F, b : F, z : F) : F ==
kernel(opKummerU, [a, b, z])
elKummerU(l : List F) : F == eKummerU(l(1), l(2), l(3))
evaluate(opKummerU, elKummerU)$BasicOperatorFunctions1(F)
eKummerUGrad_z(a : F, b : F, z : F) : F ==
((z + a - b)*kummerU(a, b, z) - kummerU(a - 1, b, z))/z
dKummerU(l : List F, t : SE) : F ==
grad3(l, t, opKummerU, eKummerUGrad_z)
setProperty(opKummerU, SPECIALDIFF, dKummerU@((List F, SE)->F)
pretend None)
-- handle LegendreP
opLegendreP := operator('legendreP)$CommonOperators
legendreP(nu, mu, z) == opLegendreP(nu, mu, z)
eLegendreP(nu : F, mu : F, z : F) : F ==
kernel(opLegendreP, [nu, mu, z])
elLegendreP(l : List F) : F == eLegendreP(l(1), l(2), l(3))
evaluate(opLegendreP, elLegendreP)$BasicOperatorFunctions1(F)
eLegendrePGrad_z(nu : F, mu : F, z : F) : F ==
(nu - mu + 1)legendreP(nu + 1, mu, z) -
(nu + 1)z*legendreP(nu, mu, z)
dLegendreP(l : List F, t : SE) : F ==
grad3(l, t, opLegendreP, eLegendrePGrad_z)
setProperty(opLegendreP, SPECIALDIFF, dLegendreP@((List F, SE)->F)
pretend None)
-- handle LegendreQ
opLegendreQ := operator('legendreQ)$CommonOperators
legendreQ(nu, mu, z) == opLegendreQ(nu, mu, z)
eLegendreQ(nu : F, mu : F, z : F) : F ==
kernel(opLegendreQ, [nu, mu, z])
elLegendreQ(l : List F) : F == eLegendreQ(l(1), l(2), l(3))
evaluate(opLegendreQ, elLegendreQ)$BasicOperatorFunctions1(F)
eLegendreQGrad_z(nu : F, mu : F, z : F) : F ==
(nu - mu + 1)legendreQ(nu + 1, mu, z) -
(nu + 1)z*legendreQ(nu, mu, z)
dLegendreQ(l : List F, t : SE) : F ==
grad3(l, t, opLegendreQ, eLegendreQGrad_z)
setProperty(opLegendreQ, SPECIALDIFF, dLegendreQ@((List F, SE)->F)
pretend None)
-- handle KelvinBei
opKelvinBei := operator('kelvinBei)$CommonOperators
kelvinBei(v, z) == opKelvinBei(v, z)
eKelvinBei(v : F, z : F) : F ==
kernel(opKelvinBei, [v, z])
elKelvinBei(l : List F) : F == eKelvinBei(l(1), l(2))
evaluate(opKelvinBei, elKelvinBei)$BasicOperatorFunctions1(F)
if F has RadicalCategory then
eKelvinBeiGrad_z(v : F, z : F) : F ==
ahalfsqrt(2::F)(kelvinBei(v + 1, z) - kelvinBer(v + 1, z)) +
v*kelvinBei(v, z)/z
dKelvinBei(l : List F, t : SE) : F ==
grad2(l, t, opKelvinBei, eKelvinBeiGrad_z)
setProperty(opKelvinBei, SPECIALDIFF, dKelvinBei@((List F, SE)->F)
pretend None)
-- handle KelvinBer
opKelvinBer := operator('kelvinBer)$CommonOperators
kelvinBer(v, z) == opKelvinBer(v, z)
eKelvinBer(v : F, z : F) : F ==
kernel(opKelvinBer, [v, z])
elKelvinBer(l : List F) : F == eKelvinBer(l(1), l(2))
evaluate(opKelvinBer, elKelvinBer)$BasicOperatorFunctions1(F)
if F has RadicalCategory then
eKelvinBerGrad_z(v : F, z : F) : F ==
ahalfsqrt(2::F)(kelvinBer(v + 1, z) + kelvinBei(v + 1, z)) +
v*kelvinBer(v, z)/z
dKelvinBer(l : List F, t : SE) : F ==
grad2(l, t, opKelvinBer, eKelvinBerGrad_z)
setProperty(opKelvinBer, SPECIALDIFF, dKelvinBer@((List F, SE)->F)
pretend None)
-- handle KelvinKei
opKelvinKei := operator('kelvinKei)$CommonOperators
kelvinKei(v, z) == opKelvinKei(v, z)
eKelvinKei(v : F, z : F) : F ==
kernel(opKelvinKei, [v, z])
elKelvinKei(l : List F) : F == eKelvinKei(l(1), l(2))
evaluate(opKelvinKei, elKelvinKei)$BasicOperatorFunctions1(F)
if F has RadicalCategory then
eKelvinKeiGrad_z(v : F, z : F) : F ==
ahalfsqrt(2::F)(kelvinKei(v + 1, z) - kelvinKer(v + 1, z)) +
v*kelvinKei(v, z)/z
dKelvinKei(l : List F, t : SE) : F ==
grad2(l, t, opKelvinKei, eKelvinKeiGrad_z)
setProperty(opKelvinKei, SPECIALDIFF, dKelvinKei@((List F, SE)->F)
pretend None)
-- handle KelvinKer
opKelvinKer := operator('kelvinKer)$CommonOperators
kelvinKer(v, z) == opKelvinKer(v, z)
eKelvinKer(v : F, z : F) : F ==
kernel(opKelvinKer, [v, z])
elKelvinKer(l : List F) : F == eKelvinKer(l(1), l(2))
evaluate(opKelvinKer, elKelvinKer)$BasicOperatorFunctions1(F)
if F has RadicalCategory then
eKelvinKerGrad_z(v : F, z : F) : F ==
ahalfsqrt(2::F)(kelvinKer(v + 1, z) + kelvinKei(v + 1, z)) +
v*kelvinKer(v, z)/z
dKelvinKer(l : List F, t : SE) : F ==
grad2(l, t, opKelvinKer, eKelvinKerGrad_z)
setProperty(opKelvinKer, SPECIALDIFF, dKelvinKer@((List F, SE)->F)
pretend None)
-- handle EllipticK
opEllipticK := operator('ellipticK)$CommonOperators
ellipticK(m) == opEllipticK(m)
eEllipticK(m : F) : F ==
kernel(opEllipticK, [m])
elEllipticK(l : List F) : F == eEllipticK(l(1))
evaluate(opEllipticK, elEllipticK)$BasicOperatorFunctions1(F)
dEllipticK(m : F) : F ==
ahalf(ellipticE(m) - (1 - m)ellipticK(m))/(m*(1 - m))
derivative(opEllipticK, dEllipticK)
-- handle one argument EllipticE
opEllipticE := operator('ellipticE)$CommonOperators
ellipticE(m) == opEllipticE(m)
eEllipticE(m : F) : F ==
kernel(opEllipticE, [m])
elEllipticE(l : List F) : F == eEllipticE(l(1))
evaluate(opEllipticE, elEllipticE)$BasicOperatorFunctions1(F)
dEllipticE(m : F) : F ==
ahalf*(ellipticE(m) - ellipticK(m))/m
derivative(opEllipticE, dEllipticE)
-- handle two argument EllipticE
opEllipticE2 := operator('ellipticE2)$CommonOperators
ellipticE(z, m) == opEllipticE2(z, m)
eEllipticE2(z : F, m : F) : F ==
z = 0 => 0
z = 1 => eEllipticE(m)
kernel(opEllipticE2, [z, m])
elEllipticE2(l : List F) : F == eEllipticE2(l(1), l(2))
evaluate(opEllipticE2, elEllipticE2)$BasicOperatorFunctions1(F)
if F has RadicalCategory then
eEllipticE2Grad_z(l : List F) : F ==
z := l(1)
m := l(2)
sqrt(1 - m*z^2)/sqrt(1 - z^2)
eEllipticE2Grad_m(l : List F) : F ==
z := l(1)
m := l(2)
ahalf*(ellipticE(z, m) - ellipticF(z, m))/m
derivative(opEllipticE2, [eEllipticE2Grad_z, eEllipticE2Grad_m])
inEllipticE2(li : List INP) : INP ==
convert cons(convert('ellipticE), li)
input(opEllipticE2, inEllipticE2@((List INP) -> INP))
-- handle EllipticF
opEllipticF := operator('ellipticF)$CommonOperators
ellipticF(z, m) == opEllipticF(z, m)
eEllipticF(z : F, m : F) : F ==
z = 0 => 0
z = 1 => ellipticK(m)
kernel(opEllipticF, [z, m])
elEllipticF(l : List F) : F == eEllipticF(l(1), l(2))
evaluate(opEllipticF, elEllipticF)$BasicOperatorFunctions1(F)
if F has RadicalCategory then
eEllipticFGrad_z(l : List F) : F ==
z := l(1)
m := l(2)
1/(sqrt(1 - mz^2)sqrt(1 - z^2))
eEllipticFGrad_m(l : List F) : F ==
z := l(1)
m := l(2)
ahalf((ellipticE(z, m) - (1 - m)ellipticF(z, m))/m -
zsqrt(1 - z^2)/sqrt(1 - mz^2))/(1 - m)
derivative(opEllipticF, [eEllipticFGrad_z, eEllipticFGrad_m])
-- handle EllipticPi
opEllipticPi := operator('ellipticPi)$CommonOperators
ellipticPi(z, n, m) == opEllipticPi(z, n, m)
eEllipticPi(z : F, n : F, m : F) : F ==
z = 0 => 0
kernel(opEllipticPi, [z, n, m])
elEllipticPi(l : List F) : F == eEllipticPi(l(1), l(2), l(3))
evaluate(opEllipticPi, elEllipticPi)$BasicOperatorFunctions1(F)
if F has RadicalCategory then
eEllipticPiGrad_z(l : List F) : F ==
z := l(1)
n := l(2)
m := l(3)
1/((1 - nz^2)sqrt(1 - mz^2)sqrt(1 - z^2))
eEllipticPiGrad_n(l : List F) : F ==
z := l(1)
n := l(2)
m := l(3)
t1 := -(n^2 - m)ellipticPi(z, n, m)/((n - 1)(n - m)n)
t2 := ellipticF(z, m)/((n - 1)n)
t3 := -ellipticE(z, m)/((n - 1)(n - m))
t4 := nzsqrt(1 - mz^2)sqrt(1 - z^2)/
((1 - nz^2)(n - 1)(n - m))
ahalf*(t1 + t2 + t3 + t4)
eEllipticPiGrad_m(l : List F) : F ==
z := l(1)
n := l(2)
m := l(3)
t1 := mzsqrt(1 - z^2)/sqrt(1 - mz^2)
t2 := (-ellipticE(z, m) + t1)/(1 - m)
ahalf(ellipticPi(z, n, m) + t2)/(n - m)
derivative(opEllipticPi, [eEllipticPiGrad_z, eEllipticPiGrad_n,
eEllipticPiGrad_m])
-- handle JacobiSn
opJacobiSn := operator('jacobiSn)$CommonOperators
jacobiSn(z, m) == opJacobiSn(z, m)
eJacobiSn(z : F, m : F) : F ==
z = 0 => 0
if is?(z, opEllipticF) then
args := argument(retract(z)@K)
m = args(2) => return args(1)
kernel(opJacobiSn, [z, m])
elJacobiSn : List F -> F
elJacobiSn(l : List F) : F == eJacobiSn(l(1), l(2))
evaluate(opJacobiSn, elJacobiSn)$BasicOperatorFunctions1(F)
jacobiGradHelper(z : F, m : F) : F ==
(z - ellipticE(jacobiSn(z, m), m)/(1 - m))/m
eJacobiSnGrad_z(l : List F) : F ==
z := l(1)
m := l(2)
jacobiCn(z, m)*jacobiDn(z, m)
eJacobiSnGrad_m(l : List F) : F ==
z := l(1)
m := l(2)
ahalf(eJacobiSnGrad_z(l)jacobiGradHelper(z, m) +
jacobiSn(z, m)*jacobiCn(z, m)^2/(1 - m))
derivative(opJacobiSn, [eJacobiSnGrad_z, eJacobiSnGrad_m])
-- handle JacobiCn
opJacobiCn := operator('jacobiCn)$CommonOperators
jacobiCn(z, m) == opJacobiCn(z, m)
eJacobiCn(z : F, m : F) : F ==
z = 0 => 1
kernel(opJacobiCn, [z, m])
elJacobiCn(l : List F) : F == eJacobiCn(l(1), l(2))
evaluate(opJacobiCn, elJacobiCn)$BasicOperatorFunctions1(F)
eJacobiCnGrad_z(l : List F) : F ==
z := l(1)
m := l(2)
-jacobiSn(z, m)*jacobiDn(z, m)
eJacobiCnGrad_m(l : List F) : F ==
z := l(1)
m := l(2)
ahalf(eJacobiCnGrad_z(l)jacobiGradHelper(z, m) -
jacobiSn(z, m)^2*jacobiCn(z, m)/(1 - m))
derivative(opJacobiCn, [eJacobiCnGrad_z, eJacobiCnGrad_m])
-- handle JacobiDn
opJacobiDn := operator('jacobiDn)$CommonOperators
jacobiDn(z, m) == opJacobiDn(z, m)
eJacobiDn(z : F, m : F) : F ==
z = 0 => 1
kernel(opJacobiDn, [z, m])
elJacobiDn(l : List F) : F == eJacobiDn(l(1), l(2))
evaluate(opJacobiDn, elJacobiDn)$BasicOperatorFunctions1(F)
eJacobiDnGrad_z(l : List F) : F ==
z := l(1)
m := l(2)
-mjacobiSn(z, m)jacobiCn(z, m)
eJacobiDnGrad_m(l : List F) : F ==
z := l(1)
m := l(2)
ahalf(eJacobiDnGrad_z(l)jacobiGradHelper(z, m) -
jacobiSn(z, m)^2*jacobiDn(z, m)/(1 - m))
derivative(opJacobiDn, [eJacobiDnGrad_z, eJacobiDnGrad_m])
-- handle JacobiZeta
opJacobiZeta := operator('jacobiZeta)$CommonOperators
jacobiZeta(z, m) == opJacobiZeta(z, m)
eJacobiZeta(z : F, m : F) : F ==
z = 0 => 0
kernel(opJacobiZeta, [z, m])
elJacobiZeta(l : List F) : F == eJacobiZeta(l(1), l(2))
evaluate(opJacobiZeta, elJacobiZeta)$BasicOperatorFunctions1(F)
eJacobiZetaGrad_z(l : List F) : F ==
z := l(1)
m := l(2)
dn := jacobiDn(z, m)
dn*dn - ellipticE(m)/ellipticK(m)
eJacobiZetaGrad_m(l : List F) : F ==
z := l(1)
m := l(2)
ek := ellipticK(m)
ee := ellipticE(m)
er := ee/ek
dn := jacobiDn(z, m)
res1 := (dndn + m - 1)jacobiZeta(z, m)
res2 := res1 + (m - 1)zdndn
res3 := res2 - mjacobiCn(z, m)jacobiDn(z, m)jacobiSn(z, m)
res4 := res3 + z(1 - m + dndn)er
ahalf(res4 - zerer)/(m*m - m)
derivative(opJacobiZeta, [eJacobiZetaGrad_z, eJacobiZetaGrad_m])
-- handle JacobiTheta
opJacobiTheta := operator('jacobiTheta)$CommonOperators
jacobiTheta(q, z) == opJacobiTheta(q, z)
eJacobiTheta(q : F, z : F) : F ==
kernel(opJacobiTheta, [q, z])
elJacobiTheta(l : List F) : F == eJacobiTheta(l(1), l(2))
evaluate(opJacobiTheta, elJacobiTheta)$BasicOperatorFunctions1(F)
-- handle LerchPhi
opLerchPhi := operator('lerchPhi)$CommonOperators
lerchPhi(z, s, a) == opLerchPhi(z, s, a)
- eLerchPhi(z : F, s : F, a : F) : F ==
- z = 0 => 1/a^s
a = 1 => polylog(s, z)/z
kernel(opLerchPhi, [z, s, a])
elLerchPhi(l : List F) : F == eLerchPhi(l(1), l(2), l(3))
evaluate(opLerchPhi, elLerchPhi)$BasicOperatorFunctions1(F)
dLerchPhi(l : List F, t : SE) : F ==
z := l(1)
s := l(2)
a := l(3)
dz := differentiate(z, t)(lerchPhi(z, s - 1, a) -
alerchPhi(z, s, a))/z
da := -differentiate(a, t)slerchPhi(z, s + 1, a)
dm := dummy
differentiate(s, t)*kernel(opdiff, [opLerchPhi [z, dm, a], dm, s])
+ dz + da
setProperty(opLerchPhi, SPECIALDIFF, dLerchPhi@((List F, SE)->F)
pretend None)
-- handle RiemannZeta
opRiemannZeta := operator('riemannZeta)$CommonOperators
riemannZeta(z) == opRiemannZeta(z)
eRiemannZeta(z : F) : F ==
kernel(opRiemannZeta, [z])
elRiemannZeta(l : List F) : F == eRiemannZeta(l(1))
evaluate(opRiemannZeta, elRiemannZeta)$BasicOperatorFunctions1(F)
-- orthogonal polynomials
charlierC(n : F, a : F, z : F) : F == opCharlierC(n, a, z)
eCharlierC(n : F, a : F, z : F) : F ==
n = 0 => 1
n = 1 => (z - a)/a
kernel(opCharlierC, [n, a, z])
elCharlierC(l : List F) : F == eCharlierC(l(1), l(2), l(3))
evaluate(opCharlierC, elCharlierC)$BasicOperatorFunctions1(F)
hermiteH(n : F, z: F) : F == opHermiteH(n, z)
eHermiteH(n : F, z: F) : F ==
n = -1 => 0
n = 0 => 1
n = 1 => (2::F)*z
kernel(opHermiteH, [n, z])
elHermiteH(l : List F) : F == eHermiteH(l(1), l(2))
evaluate(opHermiteH, elHermiteH)$BasicOperatorFunctions1(F)
eHermiteHGrad_z(n : F, z : F) : F == (2::F)nhermiteH(n - 1, z)
dHermiteH(l : List F, t : SE) : F ==
grad2(l, t, opHermiteH, eHermiteHGrad_z)
setProperty(opHermiteH, SPECIALDIFF, dHermiteH@((List F, SE)->F)
pretend None)
jacobiP(n : F, a : F, b : F, z : F) : F == opJacobiP(n, a, b, z)
eJacobiP(n : F, a : F, b : F, z : F) : F ==
n = -1 => 0
n = 0 => 1
n = 1 => ahalf(a - b) + (1 + ahalf(a + b))*z
kernel(opJacobiP, [n, a, b, z])
elJacobiP(l : List F) : F == eJacobiP(l(1), l(2), l(3), l(4))
evaluate(opJacobiP, elJacobiP)$BasicOperatorFunctions1(F)
eJacobiPGrad_z(n : F, a : F, b : F, z : F) : F ==
ahalf(a + b + n + 1)jacobiP(n - 1, a + 1, b + 1, z)
dJacobiP(l : List F, t : SE) : F ==
grad4(l, t, opJacobiP, eJacobiPGrad_z)
setProperty(opJacobiP, SPECIALDIFF, dJacobiP@((List F, SE)->F)
pretend None)
laguerreL(n : F, a : F, z : F) : F == opLaguerreL(n, a, z)
eLaguerreL(n : F, a : F, z : F) : F ==
n = -1 => 0
n = 0 => 1
n = 1 => (1 + a - z)
kernel(opLaguerreL, [n, a, z])
elLaguerreL(l : List F) : F == eLaguerreL(l(1), l(2), l(3))
evaluate(opLaguerreL, elLaguerreL)$BasicOperatorFunctions1(F)
eLaguerreLGrad_z(n : F, a : F, z : F) : F ==
laguerreL(n - 1, a + 1, z)
dLaguerreL(l : List F, t : SE) : F ==
grad3(l, t, opLaguerreL, eLaguerreLGrad_z)
setProperty(opLaguerreL, SPECIALDIFF, dLaguerreL@((List F, SE)->F)
pretend None)
meixnerM(n : F, b : F, c : F, z : F) : F == opMeixnerM(n, b, c, z)
eMeixnerM(n : F, b : F, c : F, z : F) : F ==
n = 0 => 1
n = 1 => (c - 1)z/(cb) + 1
kernel(opMeixnerM, [n, b, c, z])
elMeixnerM(l : List F) : F == eMeixnerM(l(1), l(2), l(3), l(4))
evaluate(opMeixnerM, elMeixnerM)$BasicOperatorFunctions1(F)
--
belong? op == has?(op, SPECIAL)
operator op ==
is?(op, 'abs) => opabs
is?(op, 'conjugate)=> opconjugate
is?(op, 'Gamma) => opGamma
is?(op, 'Gamma2) => opGamma2
is?(op, 'Beta) => opBeta
is?(op, 'digamma) => opdigamma
is?(op, 'polygamma)=> oppolygamma
is?(op, 'besselJ) => opBesselJ
is?(op, 'besselY) => opBesselY
is?(op, 'besselI) => opBesselI
is?(op, 'besselK) => opBesselK
is?(op, 'airyAi) => opAiryAi
is?(op, 'airyAiPrime) => opAiryAiPrime
is?(op, 'airyBi) => opAiryBi
is?(op, 'airyBiPrime) => opAiryBiPrime
is?(op, 'lambertW) => opLambertW
is?(op, 'polylog) => opPolylog
is?(op, 'weierstrassP) => opWeierstrassP
is?(op, 'weierstrassPPrime) => opWeierstrassPPrime
is?(op, 'weierstrassSigma) => opWeierstrassSigma
is?(op, 'weierstrassZeta) => opWeierstrassZeta
is?(op, 'hypergeometricF) => opHypergeometricF
is?(op, 'meijerG) => opMeijerG
-- is?(op, 'weierstrassPInverse) => opWeierstrassPInverse
is?(op, 'whittakerM) => opWhittakerM
is?(op, 'whittakerW) => opWhittakerW
is?(op, 'angerJ) => opAngerJ
is?(op, 'weberE) => opWeberE
is?(op, 'struveH) => opStruveH
is?(op, 'struveL) => opStruveL
is?(op, 'hankelH1) => opHankelH1
is?(op, 'hankelH2) => opHankelH2
is?(op, 'lommelS1) => opLommelS1
is?(op, 'lommelS2) => opLommelS2
is?(op, 'kummerM) => opKummerM
is?(op, 'kummerU) => opKummerU
is?(op, 'legendreP) => opLegendreP
is?(op, 'legendreQ) => opLegendreQ
is?(op, 'kelvinBei) => opKelvinBei
is?(op, 'kelvinBer) => opKelvinBer
is?(op, 'kelvinKei) => opKelvinKei
is?(op, 'kelvinKer) => opKelvinKer
is?(op, 'ellipticK) => opEllipticK
is?(op, 'ellipticE) => opEllipticE
is?(op, 'ellipticE2) => opEllipticE2
is?(op, 'ellipticF) => opEllipticF
is?(op, 'ellipticPi) => opEllipticPi
is?(op, 'jacobiSn) => opJacobiSn
is?(op, 'jacobiCn) => opJacobiCn
is?(op, 'jacobiDn) => opJacobiDn
is?(op, 'jacobiZeta) => opJacobiZeta
is?(op, 'jacobiTheta) => opJacobiTheta
is?(op, 'lerchPhi) => opLerchPhi
is?(op, 'riemannZeta) => opRiemannZeta
is?(op, 'charlierC) => opCharlierC
is?(op, 'hermiteH) => opHermiteH
is?(op, 'jacobiP) => opJacobiP
is?(op, 'laguerreL) => opLaguerreL
is?(op, 'meixnerM) => opMeixnerM
is?(op, '%logGamma) => op_log_gamma
is?(op, '%eis) => op_eis
is?(op, '%erfs) => op_erfs
is?(op, '%erfis) => op_erfis
error "Not a special operator"
-- Could put more unconditional special rules for other functions here
iGamma x ==
-- one? x => x
(x = 1) => x
kernel(opGamma, x)
iabs x ==
zero? x => 0
one? x => 1
is?(x, opabs) => x
is?(x, opconjugate) => kernel(opabs, argument(retract(x)@K)(1))
smaller?(x, 0) => kernel(opabs, -x)
kernel(opabs, x)
iconjugate(k:K):F ==
--output("in: ",k::OutputForm)$OutputPackage
if is?(k, opconjugate) then
x:=argument(k)(1)
else if is?(k, opabs) then
x:= k::F
else if symbolIfCan(k) case Symbol then
x:=kernel(opconjugate, k::F)
else if F has RadicalCategory and R has RetractableTo(Integer) and is?(k, 'nthRoot) then
x:= 1/nthRoot(1/iiconjugate argument(k)(1),retract(argument(k)(2))@Integer)
else -- assume holomorphic
x:=map(iiconjugate,k)
--output("out: ",x::OutputForm)$OutputPackage
return x
iiconjugate(x:F):F ==
--output("iin: ",x::OutputForm)$OutputPackage
if (s:=isPlus(x)) case List F then
--output("isPlus: ",s::OutputForm)$OutputPackage
x := reduce(_+$F,map(iiconjugate,s))
else if (s:=isTimes(x)) case List F then
--output("isTimes: ",s::OutputForm)$OutputPackage
x:= reduce(*$F,map(iiconjugate,s))
else if #(ks:List K:=kernels(x))>0 then
--output("kernels: ",ks::OutputForm)$OutputPackage
x := eval(x,ks,
map((k:Kernel F):F +-> (height(k)=0 =>k::F;iconjugate k),
ks)$ListFunctions2(Kernel F,F))
else if R has conjugate : R -> R then
x:=map(conjugate$R,numer x)::F /
map(conjugate$R,denom x)::F
--output("iout: ",x::OutputForm)$OutputPackage
return x
iBeta(x, y) == kernel(opBeta, [x, y])
idigamma x == kernel(opdigamma, x)
iiipolygamma(n, x) == kernel(oppolygamma, [n, x])
iiiBesselJ(x, y) == kernel(opBesselJ, [x, y])
iiiBesselY(x, y) == kernel(opBesselY, [x, y])
iiiBesselI(x, y) == kernel(opBesselI, [x, y])
iiiBesselK(x, y) == kernel(opBesselK, [x, y])
import from Fraction(Integer)
if F has ElementaryFunctionCategory then
iAiryAi x ==
zero?(x) => 1::F/((3::F)^twothirdsGamma(twothirds))
kernel(opAiryAi, x)
iAiryAiPrime x ==
zero?(x) => -1::F/((3::F)^athirdGamma(athird))
kernel(opAiryAiPrime, x)
iAiryBi x ==
zero?(x) => 1::F/((3::F)^asixth*Gamma(twothirds))
kernel(opAiryBi, x)
iAiryBiPrime x ==
zero?(x) => (3::F)^asixth/Gamma(athird)
kernel(opAiryBiPrime, x)
else
iAiryAi x == kernel(opAiryAi, x)
iAiryAiPrime x == kernel(opAiryAiPrime, x)
iAiryBi x == kernel(opAiryBi, x)
iAiryBiPrime x == kernel(opAiryBiPrime, x)
if F has ElementaryFunctionCategory then
iLambertW(x) ==
zero?(x) => 0
x = exp(1$F) => 1$F
x = -exp(-1$F) => -1$F
kernel(opLambertW, x)
else
iLambertW(x) ==
zero?(x) => 0
kernel(opLambertW, x)
if F has ElementaryFunctionCategory then
if F has LiouvillianFunctionCategory then
iiPolylog(s, x) ==
s = 1 => -log(1 - x)
s = 2::F => dilog(1 - x)
kernel(opPolylog, [s, x])
else
iiPolylog(s, x) ==
s = 1 => -log(1 - x)
kernel(opPolylog, [s, x])
else
iiPolylog(s, x) == kernel(opPolylog, [s, x])
iPolylog(l) == iiPolylog(first l, second l)
iWeierstrassP(g2, g3, x) == kernel(opWeierstrassP, [g2, g3, x])
iWeierstrassPPrime(g2, g3, x) == kernel(opWeierstrassPPrime, [g2, g3, x])
iWeierstrassSigma(g2, g3, x) ==
x = 0 => 0
kernel(opWeierstrassSigma, [g2, g3, x])
iWeierstrassZeta(g2, g3, x) == kernel(opWeierstrassZeta, [g2, g3, x])
-- Could put more conditional special rules for other functions here
if R has abs : R -> R then
import from Polynomial R
iiabs x ==
(r := retractIfCan(x)@Union(Fraction Polynomial R, "failed"))
case "failed" => iabs x
f := r::Fraction Polynomial R
(a := retractIfCan(numer f)@Union(R, "failed")) case "failed" or
(b := retractIfCan(denom f)@Union(R,"failed")) case "failed" => iabs x
abs(a::R)::F / abs(b::R)::F
else
if F has RadicalCategory and R has conjugate : R -> R then
iiabs x ==
(r := retractIfCan(x)@Union(R, "failed"))
case "failed" => iabs x
sqrt( (r::R*conjugate(r::R))::F)
else iiabs x == iabs x
if R has SpecialFunctionCategory then
iiGamma x ==
(r := retractIfCan(x)@Union(R,"failed")) case "failed" => iGamma x
Gamma(r::R)::F
iiBeta l ==
(r := retractIfCan(first l)@Union(R,"failed")) case "failed" or
(s := retractIfCan(second l)@Union(R,"failed")) case "failed"
=> iBeta(first l, second l)
Beta(r::R, s::R)::F
iidigamma x ==
(r := retractIfCan(x)@Union(R,"failed")) case "failed" => idigamma x
digamma(r::R)::F
iipolygamma l ==
(s := retractIfCan(first l)@Union(R,"failed")) case "failed" or
(r := retractIfCan(second l)@Union(R,"failed")) case "failed"
=> iiipolygamma(first l, second l)
polygamma(s::R, r::R)::F
iiBesselJ l ==
(r := retractIfCan(first l)@Union(R,"failed")) case "failed" or
(s := retractIfCan(second l)@Union(R,"failed")) case "failed"
=> iiiBesselJ(first l, second l)
besselJ(r::R, s::R)::F
iiBesselY l ==
(r := retractIfCan(first l)@Union(R,"failed")) case "failed" or
(s := retractIfCan(second l)@Union(R,"failed")) case "failed"
=> iiiBesselY(first l, second l)
besselY(r::R, s::R)::F
iiBesselI l ==
(r := retractIfCan(first l)@Union(R,"failed")) case "failed" or
(s := retractIfCan(second l)@Union(R,"failed")) case "failed"
=> iiiBesselI(first l, second l)
besselI(r::R, s::R)::F
iiBesselK l ==
(r := retractIfCan(first l)@Union(R,"failed")) case "failed" or
(s := retractIfCan(second l)@Union(R,"failed")) case "failed"
=> iiiBesselK(first l, second l)
besselK(r::R, s::R)::F
iiAiryAi x ==
(r := retractIfCan(x)@Union(R,"failed")) case "failed" => iAiryAi x
airyAi(r::R)::F
iiAiryAiPrime x ==
(r := retractIfCan(x)@Union(R,"failed")) case "failed" => iAiryAiPrime x
airyAiPrime(r::R)::F
iiAiryBi x ==
(r := retractIfCan(x)@Union(R,"failed")) case "failed" => iAiryBi x
airyBi(r::R)::F
iiAiryBi x ==
(r := retractIfCan(x)@Union(R,"failed")) case "failed" => iAiryBiPrime x
airyBiPrime(r::R)::F
else
if R has RetractableTo Integer then
iiGamma x ==
(r := retractIfCan(x)@Union(Integer, "failed")) case Integer
and (r::Integer >= 1) => factorial(r::Integer - 1)::F
iGamma x
else
iiGamma x == iGamma x
iiBeta l == iBeta(first l, second l)
iidigamma x == idigamma x
iipolygamma l == iiipolygamma(first l, second l)
iiBesselJ l == iiiBesselJ(first l, second l)
iiBesselY l == iiiBesselY(first l, second l)
iiBesselI l == iiiBesselI(first l, second l)
iiBesselK l == iiiBesselK(first l, second l)
iiAiryAi x == iAiryAi x
iiAiryAiPrime x == iAiryAiPrime x
iiAiryBi x == iAiryBi x
iiAiryBiPrime x == iAiryBiPrime x
iiWeierstrassP l == iWeierstrassP(first l, second l, third l)
iiWeierstrassPPrime l == iWeierstrassPPrime(first l, second l, third l)
iiWeierstrassSigma l == iWeierstrassSigma(first l, second l, third l)
iiWeierstrassZeta l == iWeierstrassZeta(first l, second l, third l)
-- Default behaviour is to build a kernel
evaluate(opGamma, iiGamma)$BasicOperatorFunctions1(F)
evaluate(opabs, iiabs)$BasicOperatorFunctions1(F)
evaluate(opconjugate, iiconjugate)$BasicOperatorFunctions1(F)
-- evaluate(opGamma2 , iiGamma2 )$BasicOperatorFunctions1(F)
evaluate(opBeta , iiBeta )$BasicOperatorFunctions1(F)
evaluate(opdigamma , iidigamma )$BasicOperatorFunctions1(F)
evaluate(oppolygamma , iipolygamma)$BasicOperatorFunctions1(F)
evaluate(opBesselJ , iiBesselJ )$BasicOperatorFunctions1(F)
evaluate(opBesselY , iiBesselY )$BasicOperatorFunctions1(F)
evaluate(opBesselI , iiBesselI )$BasicOperatorFunctions1(F)
evaluate(opBesselK , iiBesselK )$BasicOperatorFunctions1(F)
evaluate(opAiryAi , iiAiryAi )$BasicOperatorFunctions1(F)
evaluate(opAiryAiPrime, iiAiryAiPrime)$BasicOperatorFunctions1(F)
evaluate(opAiryBi , iiAiryBi )$BasicOperatorFunctions1(F)
evaluate(opAiryBiPrime, iiAiryBiPrime)$BasicOperatorFunctions1(F)
evaluate(opLambertW, iLambertW)$BasicOperatorFunctions1(F)
evaluate(opPolylog, iPolylog)$BasicOperatorFunctions1(F)
evaluate(opWeierstrassP, iiWeierstrassP)$BasicOperatorFunctions1(F)
evaluate(opWeierstrassPPrime,
iiWeierstrassPPrime)$BasicOperatorFunctions1(F)
evaluate(opWeierstrassSigma, iiWeierstrassSigma)$BasicOperatorFunctions1(F)
evaluate(opWeierstrassZeta, iiWeierstrassZeta)$BasicOperatorFunctions1(F)
evaluate(opHypergeometricF, iiHypergeometricF)$BasicOperatorFunctions1(F)
evaluate(opMeijerG, iiMeijerG)$BasicOperatorFunctions1(F)
diff1(op : OP, n : F, x : F) : F ==
dm := dummy
kernel(opdiff, [op [dm, x], dm, n])
iBesselJ(l : List F, t : SE) : F ==
n := first l; x := second l
differentiate(n, t)diff1(opBesselJ, n, x)
+ differentiate(x, t) ahalf * (besselJ (n-1, x) - besselJ (n+1, x))
iBesselY(l : List F, t : SE) : F ==
n := first l; x := second l
differentiate(n, t)diff1(opBesselY, n, x)
+ differentiate(x, t) ahalf * (besselY (n-1, x) - besselY (n+1, x))
iBesselI(l : List F, t : SE) : F ==
n := first l; x := second l
differentiate(n, t)diff1(opBesselI, n, x)
+ differentiate(x, t) ahalf * (besselI (n-1, x) + besselI (n+1, x))
iBesselK(l : List F, t : SE) : F ==
n := first l; x := second l
differentiate(n, t)diff1(opBesselK, n, x)
- differentiate(x, t) ahalf * (besselK (n-1, x) + besselK (n+1, x))
dPolylog(l : List F, t : SE) : F ==
s := first l; x := second l
differentiate(s, t)diff1(opPolylog, s, x)
+ differentiate(x, t)polylog(s-1, x)/x
ipolygamma(l : List F, x : SE) : F ==
import from List(Symbol)
member?(x, variables first l) =>
error "cannot differentiate polygamma with respect to the first argument"
n := first l; y := second l
differentiate(y, x)polygamma(n+1, y)
iBetaGrad1(l : List F) : F ==
x := first l; y := second l
Beta(x, y)(digamma x - digamma(x+y))
iBetaGrad2(l : List F) : F ==
x := first l; y := second l
Beta(x, y)*(digamma y - digamma(x+y))
if F has ElementaryFunctionCategory then
iGamma2(l : List F, t : SE) : F ==
a := first l; x := second l
differentiate(a, t)diff1(opGamma2, a, x)
- differentiate(x, t) x ^ (a - 1) * exp(-x)
setProperty(opGamma2, SPECIALDIFF, iGamma2@((List F, SE)->F)
pretend None)
inGamma2(li : List INP) : INP ==
convert cons(convert('Gamma), li)
input(opGamma2, inGamma2@((List INP) -> INP))
dLambertW(x : F) : F ==
lw := lambertW(x)
lw/(x*(1+lw))
iWeierstrassPGrad1(l : List F) : F ==
g2 := first l
g3 := second l
x := third l
delta := g2^3 - 27g3^2
wp := weierstrassP(g2, g3, x)
(weierstrassPPrime(g2, g3, x)(-9ahalfg3
weierstrassZeta(g2, g3, x) + afourthg2^2x)
- 9g3wp^2 + ahalfg2^2wp + 3ahalfg2g3)/delta
iWeierstrassPGrad2(l : List F) : F ==
g2 := first l
g3 := second l
x := third l
delta := g2^3 - 27g3^2
wp := weierstrassP(g2, g3, x)
(weierstrassPPrime(g2, g3, x)(3g2weierstrassZeta(g2, g3, x)
- 9ahalfg3x) + 6g2wp^2 - 9g3*wp-g2^2)/delta
iWeierstrassPGrad3(l : List F) : F ==
weierstrassPPrime(first l, second l, third l)
iWeierstrassPPrimeGrad1(l : List F) : F ==
g2 := first l
g3 := second l
x := third l
delta := g2^3 - 27g3^2
wp := weierstrassP(g2, g3, x)
wpp := weierstrassPPrime(g2, g3, x)
wpp2 := 6wp^2 - ahalfg2
(wpp2(-9ahalfg3weierstrassZeta(g2, g3, x) + afourthg2^2x)
+ wpp(9ahalfg3wp + afourthg2^2) - 18g3wpwpp
+ ahalfg2^2*wpp)/delta
iWeierstrassPPrimeGrad2(l : List F) : F ==
g2 := first l
g3 := second l
x := third l
delta := g2^3 - 27g3^2
wp := weierstrassP(g2, g3, x)
wpp := weierstrassPPrime(g2, g3, x)
wpp2 := 6wp^2 - ahalfg2
(wpp2(3g2weierstrassZeta(g2, g3, x) - 9ahalfg3x)
+ wpp(-3g2wp - 9ahalfg3) + 12g2wpwpp - 9g3*wpp)/delta
iWeierstrassPPrimeGrad3(l : List F) : F ==
g2 := first l
6weierstrassP(g2, second l, third l)^2 - ahalfg2
iWeierstrassSigmaGrad1(l : List F) : F ==
g2 := first l
g3 := second l
x := third l
delta := g2^3 - 27g3^2
ws := weierstrassSigma(g2, g3, x)
wz := weierstrassZeta(g2, g3, x)
wsp := wzws
wsp2 := - weierstrassP(g2, g3, x)ws + wz^2ws
afourth(-9g3wsp2 - g2^2ws
- 3afourthg2g3x^2ws + g2^2x*wsp)/delta
iWeierstrassSigmaGrad2(l : List F) : F ==
g2 := first l
g3 := second l
x := third l
delta := g2^3 - 27g3^2
ws := weierstrassSigma(g2, g3, x)
wz := weierstrassZeta(g2, g3, x)
wsp := wzws
wsp2 := - weierstrassP(g2, g3, x)ws + wz^2ws
ahalf(3g2wsp2 + 9g3ws
+ afourthg2^2x^2ws - 9g3x*wsp)/delta
iWeierstrassSigmaGrad3(l : List F) : F ==
g2 := first l
g3 := second l
x := third l
weierstrassZeta(g2, g3, x)*weierstrassSigma(g2, g3, x)
iWeierstrassZetaGrad1(l : List F) : F ==
g2 := first l
g3 := second l
x := third l
delta := g2^3 - 27g3^2
wp := weierstrassP(g2, g3, x)
(ahalfweierstrassZeta(g2, g3, x)(9g3wp + ahalfg2^2)
- ahalfg2x(ahalfg2wp+3afourthg3)
+ 9afourthg3weierstrassPPrime(g2, g3, x))/delta
iWeierstrassZetaGrad2(l : List F) : F ==
g2 := first l
g3 := second l
x := third l
delta := g2^3 - 27g3^2
wp := weierstrassP(g2, g3, x)
(-3weierstrassZeta(g2, g3, x)(g2wp + 3ahalfg3) +
ahalfx(9g3wp + ahalfg2^2)
- 3ahalfg2weierstrassPPrime(g2, g3, x))/delta
iWeierstrassZetaGrad3(l : List F) : F ==
-weierstrassP(first l, second l, third l)
OF ==> OutputForm
SEX ==> SExpression
NNI ==> NonNegativeInteger
dconjugate(lo : List OF) : OF == overbar lo.1
display(opconjugate,dconjugate)
if F has RetractableTo(Integer) then
get_int_listf : List F -> List Integer
get_int_listo : (Integer, List OF) -> List Integer
get_int_listi : (Integer, List INP) -> List Integer
get_int_listf(lf : List F) : List Integer ==
map(z +-> retract(z)@Integer, lf)$ListFunctions2(F, Integer)
replace_i(lp : List F, v : F, i : NNI) : List F ==
concat(first(lp, (i - 1)::NNI), cons(v, rest(lp, i)))
iiHypergeometricF(l) ==
n := #l
z := l(n-2)
if z = 0 then
nn := (n - 2)::NNI
pq := rest(l, nn)
pqi := get_int_listf(pq)
p := first(pqi)
q := first(rest(pqi))
p <= q + 1 => return 1
kernel(opHypergeometricF, l)
idvsum(op : BasicOperator, n : Integer, l : List F, x : Symbol) : F ==
res : F := 0
for i in 1..n for a in l repeat
dm := dummy
nl := replace_i(l, dm, i)
res := res + differentiate(a, x)*kernel(opdiff, [op nl, dm, a])
res
dvhypergeom(l : List F, x : Symbol) : F ==
n := #l
nn := (n - 2)::NNI
pq := rest(l, nn)
pqi := get_int_listf(pq)
ol := l
l := first(l, nn)
l1 := reverse(l)
z := first(l1)
p := first(pqi)
q := first(rest(pqi))
aprod := 1@F
nl := []@(List F)
for i in 1..p repeat
a := first(l)
nl := cons(a + 1, nl)
aprod := aprod a
l := rest(l)
bprod := 1@F
for i in 1..q repeat
b := first(l)
nl := cons(b + 1, nl)
bprod := bprod b
l := rest(l)
nl0 := reverse!(nl)
nl1 := cons(z, pq)
nl := concat(nl0, nl1)
aprod := aprod/bprod
idvsum(opHypergeometricF, nn - 1, ol, x) +
differentiate(z, x)aprodopHypergeometricF(nl)
add_pairs_to_list(lp : List List F, l : List F) : List F ==
for p in lp repeat
#p ~= 2 => error "not a list of pairs"
l := cons(p(2), cons(p(1), l))
l
dvmeijer(l : List F, x : Symbol) : F ==
n := #l
nn := (n - 4)::NNI
l0 := l
nl := rest(l, nn)
nli := get_int_listf(nl)
l := first(l, nn)
l1 := reverse(l)
z := first(l1)
n1 := first(nli)
n2 := nli(2)
a := first l
sign : F := 1
if n1 > 0 or n2 > 0 then
na := a - 1
if n1 = 0 then sign := -1
l2 := cons(na, rest l)
else
na := a
if nli(3) > 0 then sign := -1
l2 := cons(a + 1, rest l)
nm : F := opMeijerG(concat(l2, nl))
om : F := opMeijerG(l0)
idvsum(opMeijerG, nn - 1, l0, x) +
differentiate(z, x)(signnm + na*om)/z
get_if_list(n : Integer, lf : List INP) : List List INP ==
a := []@(List INP)
for i in 1..n repeat
a := cons(first(lf), a)
lf := rest(lf)
a := cons(convert('construct), reverse!(a))
[a, lf]
get_if_lists(ln : List Integer, lf : List INP) : List List INP ==
rl := []@(List List INP)
for n in ln repeat
al := get_if_list(n, lf)
rl := cons(first(al), rl)
lf := first(rest(al))
rl := reverse!(rl)
cons(lf, rl)
get_int_listi(n : Integer, lo : List INP) : List Integer ==
n0 := (#lo - n)::NNI
lo := rest(lo, n0)
rl := []@(List Integer)
for i in 1..n repeat
p := integer(first(lo) pretend SEX)$SEX
rl := cons(p, rl)
lo := rest(lo)
rl := reverse!(rl)
rl
get_of_list(n : Integer, lo : List OF) : List List OF ==
a := []@(List OF)
for i in 1..n repeat
a := cons(first(lo), a)
lo := rest(lo)
a := reverse!(a)
[a, lo]
get_of_lists(ln : List Integer, lo : List OF) : List List OF ==
rl := []@(List List OF)
for n in ln repeat
al := get_of_list(n, lo)
rl := cons(first(al), rl)
lo := first(rest(al))
rl := reverse!(rl)
cons(lo, rl)
get_int_listo(n : Integer, lo : List OF) : List Integer ==
n0 := (#lo - n)::NNI
lo := rest(lo, n0)
rl := []@(List Integer)
for i in 1..n repeat
p := integer(first(lo) pretend SEX)$SEX
rl := cons(p, rl)
lo := rest(lo)
rl := reverse!(rl)
rl
dhyper0(op : OF, lo : List OF) : OF ==
n0 := (#lo - 2)::NNI
pql := get_int_listo(2, lo)
lo := first(lo, n0)
al := get_of_lists(pql, lo)
lo := first(al)
al := rest(al)
a := first al
b := first(rest(al))
z := first(lo)
prefix(op, [bracket a, bracket b, z])
dhyper(lo : List OF) : OF ==
dhyper0(hypergeometricF, lo)
ddhyper(lo : List OF) : OF ==
dhyper0(first lo, rest lo)
dmeijer0(op : OF, lo : List OF) : OF ==
n0 := (#lo - 4)::NNI
nl := get_int_listo(4, lo)
lo := first(lo, n0)
al := get_of_lists(nl, lo)
lo := first(al)
al := rest(al)
z := first(lo)
prefix(op, concat(
map(bracket, al)$ListFunctions2(List OF, OF), [z]))
dmeijer(lo : List OF) : OF ==
dmeijer0('meijerG::OF, lo)
ddmeijer(lo : List OF) : OF ==
dmeijer0(first lo, rest lo)
setProperty(opHypergeometricF, '%diffDisp,
ddhyper@(List OF -> OF) pretend None)
setProperty(opMeijerG, '%diffDisp,
ddmeijer@(List OF -> OF) pretend None)
display(opHypergeometricF, dhyper)
display(opMeijerG, dmeijer)
setProperty(opHypergeometricF, SPECIALDIFF,
dvhypergeom@((List F, Symbol)->F) pretend None)
setProperty(opMeijerG, SPECIALDIFF, dvmeijer@((List F, Symbol)->F)
pretend None)
inhyper(lf : List INP) : INP ==
pqi := get_int_listi(2, lf)
al := get_if_lists(pqi, lf)
lf := first(al)
al := rest(al)
a := first al
ai : INP := convert(a)
b := first(rest(al))
bi : INP := convert(b)
zi := first(lf)
li : List INP := [convert('hypergeometricF), ai, bi, zi]
convert(li)
input(opHypergeometricF, inhyper@((List INP) -> INP))
inmeijer(lf : List INP) : INP ==
pqi := get_int_listi(4, lf)
al := get_if_lists(pqi, lf)
lf := first(al)
al := rest(al)
a := first al
ai : INP := convert(a)
al := rest(al)
b := first(al)
bi : INP := convert(b)
al := rest(al)
c := first(al)
ci : INP := convert(c)
al := rest(al)
d := first(al)
di : INP := convert(d)
zi := first(lf)
li : List INP := [convert('meijerG), ai, bi, ci, di, zi]
convert(li)
input(opMeijerG, inmeijer@((List INP) -> INP))
else
iiHypergeometricF(l) == kernel(opHypergeometricF, l)
iiMeijerG(l) == kernel(opMeijerG, l)
d_eis(x : F) : F == -kernel(op_eis, x) + 1/x
if F has TranscendentalFunctionCategory
and F has RadicalCategory then
d_erfs(x : F) : F == 2xkernel(op_erfs, x) - 2::F/sqrt(pi())
d_erfis(x : F) : F == -2xkernel(op_erfis, x) + 2::F/sqrt(pi())
derivative(op_erfs, d_erfs)
derivative(op_erfis, d_erfis)
derivative(opabs, (x : F) : F +-> conjugate(x)inv(2abs(x)))
derivative(opconjugate, (x : F) : F +-> 0)
derivative(opGamma, (x : F) : F +-> digamma(x)*Gamma(x))
derivative(op_log_gamma, (x : F) : F +-> digamma(x))
derivative(opBeta, [iBetaGrad1, iBetaGrad2])
derivative(opdigamma, (x : F) : F +-> polygamma(1, x))
derivative(op_eis, d_eis)
derivative(opAiryAi, (x : F) : F +-> airyAiPrime(x))
derivative(opAiryAiPrime, (x : F) : F +-> xairyAi(x))
derivative(opAiryBi, (x : F) : F +-> airyBiPrime(x))
derivative(opAiryBiPrime, (x : F) : F +-> xairyBi(x))
derivative(opLambertW, dLambertW)
derivative(opWeierstrassP, [iWeierstrassPGrad1, iWeierstrassPGrad2,
iWeierstrassPGrad3])
derivative(opWeierstrassPPrime, [iWeierstrassPPrimeGrad1,
iWeierstrassPPrimeGrad2, iWeierstrassPPrimeGrad3])
derivative(opWeierstrassSigma, [iWeierstrassSigmaGrad1,
iWeierstrassSigmaGrad2, iWeierstrassSigmaGrad3])
derivative(opWeierstrassZeta, [iWeierstrassZetaGrad1,
iWeierstrassZetaGrad2, iWeierstrassZetaGrad3])
setProperty(oppolygamma, SPECIALDIFF, ipolygamma@((List F, SE)->F)
pretend None)
setProperty(opBesselJ, SPECIALDIFF, iBesselJ@((List F, SE)->F)
pretend None)
setProperty(opBesselY, SPECIALDIFF, iBesselY@((List F, SE)->F)
pretend None)
setProperty(opBesselI, SPECIALDIFF, iBesselI@((List F, SE)->F)
pretend None)
setProperty(opBesselK, SPECIALDIFF, iBesselK@((List F, SE)->F)
pretend None)
setProperty(opPolylog, SPECIALDIFF, dPolylog@((List F, SE)->F)
pretend None)</spad>
Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1600462418712884080-25px001.spad
using old system compiler.
FSPECX abbreviates package FunctionalSpecialFunction
Your user access level is compiler and this command is therefore not
available. See the )set userlevel command for more information.
------------------------------------------------------------------------
initializing NRLIB FSPECX for FunctionalSpecialFunction
compiling into NRLIB FSPECX
****** Domain: R already in scope
processing macro definition INP ==> InputForm
processing macro definition SPECIALINPUT ==> QUOTE %specialInput
compiling exported abs : F -> F
Time: 0.04 SEC.
compiling exported conjugate : F -> F
Time: 0 SEC.
compiling exported Gamma : F -> F
Time: 0 SEC.
compiling exported Gamma : (F,F) -> F
Time: 0 SEC.
compiling exported Beta : (F,F) -> F
Time: 0 SEC.
compiling exported digamma : F -> F
Time: 0 SEC.
compiling exported polygamma : (F,F) -> F
Time: 0 SEC.
compiling exported besselJ : (F,F) -> F
Time: 0 SEC.
compiling exported besselY : (F,F) -> F
Time: 0 SEC.
compiling exported besselI : (F,F) -> F
Time: 0 SEC.
compiling exported besselK : (F,F) -> F
Time: 0 SEC.
compiling exported airyAi : F -> F
Time: 0 SEC.
compiling exported airyAiPrime : F -> F
Time: 0 SEC.
compiling exported airyBi : F -> F
Time: 0 SEC.
compiling exported airyBiPrime : F -> F
Time: 0 SEC.
compiling exported lambertW : F -> F
Time: 0 SEC.
compiling exported polylog : (F,F) -> F
Time: 0 SEC.
compiling exported weierstrassP : (F,F,F) -> F
Time: 0 SEC.
compiling exported weierstrassPPrime : (F,F,F) -> F
Time: 0 SEC.
compiling exported weierstrassSigma : (F,F,F) -> F
Time: 0 SEC.
compiling exported weierstrassZeta : (F,F,F) -> F
Time: 0 SEC.
****** Domain: F already in scope
augmenting F: (RetractableTo (Integer))
compiling exported hypergeometricF : (List F,List F,F) -> F
Time: 0.03 SEC.
compiling exported meijerG : (List F,List F,List F,List F,F) -> F
Time: 0.01 SEC.
importing List Kernel F
processing macro definition dummy ==> ::((Sel (Symbol) new),F)
compiling local grad2 : (List F,Symbol,BasicOperator,(F,F) -> F) -> F
Time: 0.04 SEC.
compiling local grad3 : (List F,Symbol,BasicOperator,(F,F,F) -> F) -> F
Time: 0.03 SEC.
compiling local grad4 : (List F,Symbol,BasicOperator,(F,F,F,F) -> F) -> F
Time: 0.02 SEC.
compiling exported whittakerM : (F,F,F) -> F
Time: 0 SEC.
compiling local eWhittakerM : (F,F,F) -> F
Time: 0 SEC.
compiling local elWhittakerM : List F -> F
Time: 0 SEC.
compiling local eWhittakerMGrad_z : (F,F,F) -> F
Time: 0.03 SEC.
compiling local dWhittakerM : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported whittakerW : (F,F,F) -> F
Time: 0 SEC.
compiling local eWhittakerW : (F,F,F) -> F
Time: 0 SEC.
compiling local elWhittakerW : List F -> F
Time: 0 SEC.
compiling local eWhittakerWGrad_z : (F,F,F) -> F
Time: 0 SEC.
compiling local dWhittakerW : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported angerJ : (F,F) -> F
Time: 0 SEC.
****** Domain: F already in scope
augmenting F: (TranscendentalFunctionCategory)
compiling local eAngerJ : (F,F) -> F
Time: 0 SEC.
compiling local elAngerJ : List F -> F
Time: 0 SEC.
compiling local eAngerJGrad_z : (F,F) -> F
Time: 0 SEC.
compiling local dAngerJ : (List F,Symbol) -> F
Time: 0 SEC.
compiling local eeAngerJ : List F -> F
Time: 0 SEC.
compiling exported weberE : (F,F) -> F
Time: 0 SEC.
****** Domain: F already in scope
augmenting F: (TranscendentalFunctionCategory)
compiling local eWeberE : (F,F) -> F
Time: 0.21 SEC.
compiling local elWeberE : List F -> F
Time: 0 SEC.
compiling local eWeberEGrad_z : (F,F) -> F
Time: 0.01 SEC.
compiling local dWeberE : (List F,Symbol) -> F
Time: 0 SEC.
compiling local eeWeberE : List F -> F
Time: 0 SEC.
compiling exported struveH : (F,F) -> F
Time: 0 SEC.
compiling local eStruveH : (F,F) -> F
Time: 0 SEC.
compiling local elStruveH : List F -> F
Time: 0 SEC.
**** Domain: F already in scope
augmenting F: (TranscendentalFunctionCategory)
**** Domain: F already in scope
augmenting F: (RadicalCategory)
compiling local eStruveHGrad_z : (F,F) -> F
Time: 0 SEC.
compiling local dStruveH : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported struveL : (F,F) -> F
Time: 0 SEC.
compiling local eStruveL : (F,F) -> F
Time: 0 SEC.
compiling local elStruveL : List F -> F
Time: 0 SEC.
**** Domain: F already in scope
augmenting F: (TranscendentalFunctionCategory)
**** Domain: F already in scope
augmenting F: (RadicalCategory)
compiling local eStruveLGrad_z : (F,F) -> F
Time: 0 SEC.
compiling local dStruveL : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported hankelH1 : (F,F) -> F
Time: 0 SEC.
compiling local eHankelH1 : (F,F) -> F
Time: 0 SEC.
compiling local elHankelH1 : List F -> F
Time: 0 SEC.
compiling local eHankelH1Grad_z : (F,F) -> F
Time: 0 SEC.
compiling local dHankelH1 : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported hankelH2 : (F,F) -> F
Time: 0 SEC.
compiling local eHankelH2 : (F,F) -> F
Time: 0 SEC.
compiling local elHankelH2 : List F -> F
Time: 0 SEC.
compiling local eHankelH2Grad_z : (F,F) -> F
Time: 0 SEC.
compiling local dHankelH2 : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported lommelS1 : (F,F,F) -> F
Time: 0 SEC.
compiling local eLommelS1 : (F,F,F) -> F
Time: 0 SEC.
compiling local elLommelS1 : List F -> F
Time: 0 SEC.
compiling local eLommelS1Grad_z : (F,F,F) -> F
Time: 0 SEC.
compiling local dLommelS1 : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported lommelS2 : (F,F,F) -> F
Time: 0 SEC.
compiling local eLommelS2 : (F,F,F) -> F
Time: 0 SEC.
compiling local elLommelS2 : List F -> F
Time: 0 SEC.
compiling local eLommelS2Grad_z : (F,F,F) -> F
Time: 0 SEC.
compiling local dLommelS2 : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported kummerM : (F,F,F) -> F
Time: 0 SEC.
compiling local eKummerM : (F,F,F) -> F
Time: 0 SEC.
compiling local elKummerM : List F -> F
Time: 0 SEC.
compiling local eKummerMGrad_z : (F,F,F) -> F
Time: 0.04 SEC.
compiling local dKummerM : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported kummerU : (F,F,F) -> F
Time: 0 SEC.
compiling local eKummerU : (F,F,F) -> F
Time: 0 SEC.
compiling local elKummerU : List F -> F
Time: 0 SEC.
compiling local eKummerUGrad_z : (F,F,F) -> F
Time: 0.03 SEC.
compiling local dKummerU : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported legendreP : (F,F,F) -> F
Time: 0 SEC.
compiling local eLegendreP : (F,F,F) -> F
Time: 0 SEC.
compiling local elLegendreP : List F -> F
Time: 0 SEC.
compiling local eLegendrePGrad_z : (F,F,F) -> F
Time: 0.04 SEC.
compiling local dLegendreP : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported legendreQ : (F,F,F) -> F
Time: 0 SEC.
compiling local eLegendreQ : (F,F,F) -> F
Time: 0 SEC.
compiling local elLegendreQ : List F -> F
Time: 0 SEC.
compiling local eLegendreQGrad_z : (F,F,F) -> F
Time: 0.04 SEC.
compiling local dLegendreQ : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported kelvinBei : (F,F) -> F
Time: 0 SEC.
compiling local eKelvinBei : (F,F) -> F
Time: 0 SEC.
compiling local elKelvinBei : List F -> F
Time: 0 SEC.
****** Domain: F already in scope
augmenting F: (RadicalCategory)
compiling local eKelvinBeiGrad_z : (F,F) -> F
Time: 0 SEC.
compiling local dKelvinBei : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported kelvinBer : (F,F) -> F
Time: 0 SEC.
compiling local eKelvinBer : (F,F) -> F
Time: 0 SEC.
compiling local elKelvinBer : List F -> F
Time: 0 SEC.
****** Domain: F already in scope
augmenting F: (RadicalCategory)
compiling local eKelvinBerGrad_z : (F,F) -> F
Time: 0.03 SEC.
compiling local dKelvinBer : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported kelvinKei : (F,F) -> F
Time: 0 SEC.
compiling local eKelvinKei : (F,F) -> F
Time: 0 SEC.
compiling local elKelvinKei : List F -> F
Time: 0 SEC.
****** Domain: F already in scope
augmenting F: (RadicalCategory)
compiling local eKelvinKeiGrad_z : (F,F) -> F
Time: 0.01 SEC.
compiling local dKelvinKei : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported kelvinKer : (F,F) -> F
Time: 0 SEC.
compiling local eKelvinKer : (F,F) -> F
Time: 0 SEC.
compiling local elKelvinKer : List F -> F
Time: 0 SEC.
****** Domain: F already in scope
augmenting F: (RadicalCategory)
compiling local eKelvinKerGrad_z : (F,F) -> F
Time: 0.02 SEC.
compiling local dKelvinKer : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported ellipticK : F -> F
Time: 0 SEC.
compiling local eEllipticK : F -> F
Time: 0 SEC.
compiling local elEllipticK : List F -> F
Time: 0 SEC.
compiling local dEllipticK : F -> F
Time: 0.03 SEC.
compiling exported ellipticE : F -> F
Time: 0 SEC.
compiling local eEllipticE : F -> F
Time: 0 SEC.
compiling local elEllipticE : List F -> F
Time: 0 SEC.
compiling local dEllipticE : F -> F
Time: 0 SEC.
compiling exported ellipticE : (F,F) -> F
Time: 0 SEC.
compiling local eEllipticE2 : (F,F) -> F
Time: 0 SEC.
compiling local elEllipticE2 : List F -> F
Time: 0 SEC.
****** Domain: F already in scope
augmenting F: (RadicalCategory)
compiling local eEllipticE2Grad_z : List F -> F
Time: 0 SEC.
compiling local eEllipticE2Grad_m : List F -> F
Time: 0 SEC.
compiling local inEllipticE2 : List InputForm -> InputForm
Time: 0 SEC.
compiling exported ellipticF : (F,F) -> F
Time: 0 SEC.
compiling local eEllipticF : (F,F) -> F
Time: 0 SEC.
compiling local elEllipticF : List F -> F
Time: 0 SEC.
****** Domain: F already in scope
augmenting F: (RadicalCategory)
compiling local eEllipticFGrad_z : List F -> F
Time: 0.10 SEC.
compiling local eEllipticFGrad_m : List F -> F
Time: 0.18 SEC.
compiling exported ellipticPi : (F,F,F) -> F
Time: 0 SEC.
compiling local eEllipticPi : (F,F,F) -> F
Time: 0 SEC.
compiling local elEllipticPi : List F -> F
Time: 0 SEC.
****** Domain: F already in scope
augmenting F: (RadicalCategory)
compiling local eEllipticPiGrad_z : List F -> F
Time: 1.63 SEC.
compiling local eEllipticPiGrad_n : List F -> F
Time: 0.94 SEC.
compiling local eEllipticPiGrad_m : List F -> F
Time: 0.04 SEC.
compiling exported jacobiSn : (F,F) -> F
Time: 0 SEC.
compiling local eJacobiSn : (F,F) -> F
Time: 0 SEC.
compiling local elJacobiSn : List F -> F
Time: 0 SEC.
compiling local jacobiGradHelper : (F,F) -> F
Time: 0 SEC.
compiling local eJacobiSnGrad_z : List F -> F
Time: 0 SEC.
compiling local eJacobiSnGrad_m : List F -> F
Time: 0.07 SEC.
compiling exported jacobiCn : (F,F) -> F
Time: 0 SEC.
compiling local eJacobiCn : (F,F) -> F
Time: 0 SEC.
compiling local elJacobiCn : List F -> F
Time: 0 SEC.
compiling local eJacobiCnGrad_z : List F -> F
Time: 0 SEC.
compiling local eJacobiCnGrad_m : List F -> F
Time: 0.17 SEC.
compiling exported jacobiDn : (F,F) -> F
Time: 0 SEC.
compiling local eJacobiDn : (F,F) -> F
Time: 0 SEC.
compiling local elJacobiDn : List F -> F
Time: 0 SEC.
compiling local eJacobiDnGrad_z : List F -> F
Time: 0 SEC.
compiling local eJacobiDnGrad_m : List F -> F
Time: 0.17 SEC.
compiling exported jacobiZeta : (F,F) -> F
Time: 0 SEC.
compiling local eJacobiZeta : (F,F) -> F
Time: 0 SEC.
compiling local elJacobiZeta : List F -> F
Time: 0 SEC.
compiling local eJacobiZetaGrad_z : List F -> F
Time: 0 SEC.
compiling local eJacobiZetaGrad_m : List F -> F
Time: 0.67 SEC.
compiling exported jacobiTheta : (F,F) -> F
Time: 0 SEC.
compiling local eJacobiTheta : (F,F) -> F
Time: 0 SEC.
compiling local elJacobiTheta : List F -> F
Time: 0 SEC.
compiling exported lerchPhi : (F,F,F) -> F
Time: 0 SEC.
compiling local eLerchPhi : (F,F,F) -> F
Time: 0 SEC.
compiling local elLerchPhi : List F -> F
Time: 0 SEC.
compiling local dLerchPhi : (List F,Symbol) -> F
Time: 0.45 SEC.
compiling exported riemannZeta : F -> F
Time: 0 SEC.
compiling local eRiemannZeta : F -> F
Time: 0 SEC.
compiling local elRiemannZeta : List F -> F
Time: 0 SEC.
compiling exported charlierC : (F,F,F) -> F
Time: 0 SEC.
compiling local eCharlierC : (F,F,F) -> F
Time: 0 SEC.
compiling local elCharlierC : List F -> F
Time: 0 SEC.
compiling exported hermiteH : (F,F) -> F
Time: 0 SEC.
compiling local eHermiteH : (F,F) -> F
Time: 0 SEC.
compiling local elHermiteH : List F -> F
Time: 0 SEC.
compiling local eHermiteHGrad_z : (F,F) -> F
Time: 0.01 SEC.
compiling local dHermiteH : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported jacobiP : (F,F,F,F) -> F
Time: 0 SEC.
compiling local eJacobiP : (F,F,F,F) -> F
Time: 0.03 SEC.
compiling local elJacobiP : List F -> F
Time: 0 SEC.
compiling local eJacobiPGrad_z : (F,F,F,F) -> F
Time: 0.09 SEC.
compiling local dJacobiP : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported laguerreL : (F,F,F) -> F
Time: 0 SEC.
compiling local eLaguerreL : (F,F,F) -> F
Time: 0 SEC.
compiling local elLaguerreL : List F -> F
Time: 0 SEC.
compiling local eLaguerreLGrad_z : (F,F,F) -> F
Time: 0 SEC.
compiling local dLaguerreL : (List F,Symbol) -> F
Time: 0 SEC.
compiling exported meixnerM : (F,F,F,F) -> F
Time: 0 SEC.
compiling local eMeixnerM : (F,F,F,F) -> F
Time: 0.02 SEC.
compiling local elMeixnerM : List F -> F
Time: 0 SEC.
compiling exported belong? : BasicOperator -> Boolean
Time: 0 SEC.
compiling exported operator : BasicOperator -> BasicOperator
Time: 0.01 SEC.
compiling local iGamma : F -> F
Time: 0 SEC.
compiling local iabs : F -> F
Time: 0 SEC.
compiling local iconjugate : Kernel F -> F
**** Domain: F already in scope
augmenting F: (RadicalCategory)
**** Domain: R already in scope
augmenting R: (RetractableTo (Integer))
Time: 0 SEC.
compiling exported iiconjugate : F -> F
augmenting R: (SIGNATURE R conjugate (R R))
Time: 0.01 SEC.
compiling local iBeta : (F,F) -> F
Time: 0 SEC.
compiling local idigamma : F -> F
Time: 0 SEC.
compiling local iiipolygamma : (F,F) -> F
Time: 0 SEC.
compiling local iiiBesselJ : (F,F) -> F
Time: 0 SEC.
compiling local iiiBesselY : (F,F) -> F
Time: 0 SEC.
compiling local iiiBesselI : (F,F) -> F
Time: 0 SEC.
compiling local iiiBesselK : (F,F) -> F
Time: 0 SEC.
importing Fraction Integer
****** Domain: F already in scope
augmenting F: (ElementaryFunctionCategory)
compiling exported iAiryAi : F -> F
Time: 0 SEC.
compiling exported iAiryAiPrime : F -> F
Time: 0 SEC.
compiling exported iAiryBi : F -> F
Time: 0 SEC.
compiling exported iAiryBiPrime : F -> F
Time: 0 SEC.
compiling exported iAiryAi : F -> F
Time: 0 SEC.
compiling exported iAiryAiPrime : F -> F
Time: 0 SEC.
compiling exported iAiryBi : F -> F
Time: 0 SEC.
compiling exported iAiryBiPrime : F -> F
Time: 0 SEC.
****** Domain: F already in scope
augmenting F: (ElementaryFunctionCategory)
compiling exported iLambertW : F -> F
Time: 0 SEC.
compiling exported iLambertW : F -> F
Time: 0 SEC.
**** Domain: F already in scope
augmenting F: (ElementaryFunctionCategory)
**** Domain: F already in scope
augmenting F: (LiouvillianFunctionCategory)
compiling exported iiPolylog : (F,F) -> F
Time: 0 SEC.
compiling exported iiPolylog : (F,F) -> F
Time: 0 SEC.
compiling exported iiPolylog : (F,F) -> F
Time: 0 SEC.
compiling local iPolylog : List F -> F
Time: 0 SEC.
compiling local iWeierstrassP : (F,F,F) -> F
Time: 0 SEC.
compiling local iWeierstrassPPrime : (F,F,F) -> F
Time: 0 SEC.
compiling local iWeierstrassSigma : (F,F,F) -> F
Time: 0 SEC.
compiling local iWeierstrassZeta : (F,F,F) -> F
Time: 0 SEC.
augmenting R: (SIGNATURE R abs (R R))
importing Polynomial R
compiling exported iiabs : F -> F
Time: 0 SEC.
****** Domain: F already in scope
augmenting F: (RadicalCategory)
augmenting R: (SIGNATURE R conjugate (R R))
compiling exported iiabs : F -> F
Time: 0 SEC.
compiling exported iiabs : F -> F
Time: 0 SEC.
compiling exported iiabs : F -> F
Time: 0 SEC.
****** Domain: R already in scope
augmenting R: (SpecialFunctionCategory)
compiling exported iiGamma : F -> F
Time: 0 SEC.
compiling exported iiBeta : List F -> F
Time: 0 SEC.
compiling exported iidigamma : F -> F
Time: 0 SEC.
compiling exported iipolygamma : List F -> F
Time: 0 SEC.
compiling exported iiBesselJ : List F -> F
Time: 0 SEC.
compiling exported iiBesselY : List F -> F
Time: 0 SEC.
compiling exported iiBesselI : List F -> F
Time: 0 SEC.
compiling exported iiBesselK : List F -> F
Time: 0 SEC.
compiling exported iiAiryAi : F -> F
Time: 0 SEC.
compiling exported iiAiryAiPrime : F -> F
Time: 0 SEC.
compiling exported iiAiryBi : F -> F
Time: 0 SEC.
compiling exported iiAiryBi : F -> F
Time: 0 SEC.
****** Domain: R already in scope
augmenting R: (RetractableTo (Integer))
compiling exported iiGamma : F -> F
Time: 0 SEC.
compiling exported iiGamma : F -> F
Time: 0 SEC.
compiling exported iiBeta : List F -> F
Time: 0 SEC.
compiling exported iidigamma : F -> F
Time: 0 SEC.
compiling exported iipolygamma : List F -> F
Time: 0 SEC.
compiling exported iiBesselJ : List F -> F
Time: 0 SEC.
compiling exported iiBesselY : List F -> F
Time: 0 SEC.
compiling exported iiBesselI : List F -> F
Time: 0 SEC.
compiling exported iiBesselK : List F -> F
Time: 0 SEC.
compiling exported iiAiryAi : F -> F
Time: 0 SEC.
compiling exported iiAiryAiPrime : F -> F
Time: 0 SEC.
compiling exported iiAiryBi : F -> F
Time: 0 SEC.
compiling exported iiAiryBiPrime : F -> F
Time: 0 SEC.
compiling local iiWeierstrassP : List F -> F
Time: 0 SEC.
compiling local iiWeierstrassPPrime : List F -> F
Time: 0 SEC.
compiling local iiWeierstrassSigma : List F -> F
Time: 0 SEC.
compiling local iiWeierstrassZeta : List F -> F
Time: 0 SEC.
compiling local diff1 : (BasicOperator,F,F) -> F
Time: 0.32 SEC.
compiling local iBesselJ : (List F,Symbol) -> F
Time: 0.03 SEC.
compiling local iBesselY : (List F,Symbol) -> F
Time: 0.04 SEC.
compiling local iBesselI : (List F,Symbol) -> F
Time: 0.03 SEC.
compiling local iBesselK : (List F,Symbol) -> F
Time: 0.04 SEC.
compiling local dPolylog : (List F,Symbol) -> F
Time: 0.02 SEC.
compiling local ipolygamma : (List F,Symbol) -> F
Time: 0 SEC.
compiling local iBetaGrad1 : List F -> F
Time: 0 SEC.
compiling local iBetaGrad2 : List F -> F
Time: 0 SEC.
****** Domain: F already in scope
augmenting F: (ElementaryFunctionCategory)
compiling local iGamma2 : (List F,Symbol) -> F
Time: 0.04 SEC.
compiling local inGamma2 : List InputForm -> InputForm
Time: 0 SEC.
compiling local dLambertW : F -> F
Time: 0 SEC.
compiling local iWeierstrassPGrad1 : List F -> F
Time: 104.52 SEC.
compiling local iWeierstrassPGrad2 : List F -> F
Time: 9.71 SEC.
compiling local iWeierstrassPGrad3 : List F -> F
Time: 0 SEC.
compiling local iWeierstrassPPrimeGrad1 : List F -> F
Time: 126.21 SEC.
compiling local iWeierstrassPPrimeGrad2 : List F -> F
Time: 17.52 SEC.
compiling local iWeierstrassPPrimeGrad3 : List F -> F
Time: 0 SEC.
compiling local iWeierstrassSigmaGrad1 : List F -> F
Welcome to LDB, a low-level debugger for the Lisp runtime environment.
(GC in progress, oldspace=3, newspace=4)
ldb>
Some or all expressions may not have rendered properly,
because Latex returned the following error:
! Missing $ inserted.
<inserted text>
$
l.137
Overfull \hbox (99.52426pt too wide) in paragraph at lines 126--137
\OML/cmm/m/it/12 ManuelBronstein \OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 DateCreate
d \OT1/cmr/m/n/12 : 18\OML/cmm/m/it/12 Apr\OT1/cmr/m/n/12 1989 + +\OML/cmm/m/it
/12 DateLastUpdated \OT1/cmr/m/n/12 : 4\OML/cmm/m/it/12 October\OT1/cmr/m/n/12
1993 + +\OML/cmm/m/it/12 Description \OT1/cmr/m/n/12 : \OML/cmm/m/it/12 Provide
ssomespecialfunctionsoveranintegraldomain: \OT1/cmr/m/n/12 +
Overfull \hbox (40.15955pt too wide) in paragraph at lines 126--137
\OT1/cmr/m/n/12 +\OML/cmm/m/it/12 Keywords \OT1/cmr/m/n/12 : \OML/cmm/m/it/12 s
pecial; function:FunctionalSpecialFunction\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 R;
F\OT1/cmr/m/n/12 ) : \OML/cmm/m/it/12 Exports \OT1/cmr/m/n/12 == \OML/cmm/m/it/
12 ImplementationwhereR \OT1/cmr/m/n/12 : \OML/cmm/m/it/12 Join\OT1/cmr/m/n/12
(\OML/cmm/m/it/12 Comparable; IntegralDomain\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 F
\OT1/cmr/m/n/12 :
Missing $ inserted.
<inserted text>
$
l.248 ++ integrate(1/sqrt((1-t^
2)(1-mt^2)), t = 0..1)}
Extra }, or forgotten $.
l.248 ...ate(1/sqrt((1-t^2)(1-mt^2)), t = 0..1)}
Missing $ inserted.
<inserted text>
$
l.346
Overfull \hbox (45.2345pt too wide) in paragraph at lines 143--346
[]\OML/cmm/m/it/12 ellipticE \OT1/cmr/m/n/12 : (\OML/cmm/m/it/12 F; F\OT1/cmr/m
/n/12 )\OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 > F \OT1/cmr/m/n/12 + +\OML/cmm/m/
it/12 ellipticE\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 z; m\OT1/cmr/m/n/12 )\OML/cmm/
m/it/12 istheincompleteellipticintegralofthe \OT1/cmr/m/n/12 +
Overfull \hbox (133.79897pt too wide) in paragraph at lines 143--346
\OT1/cmr/m/n/12 +\OML/cmm/m/it/12 secondkind \OT1/cmr/m/n/12 : []\OML/cmm/m/it/
12 ellipticF \OT1/cmr/m/n/12 : (\OML/cmm/m/it/12 F; F\OT1/cmr/m/n/12 )\OMS/cmsy
/m/n/12 ^^@ \OML/cmm/m/it/12 > F \OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 ellipticF\
OT1/cmr/m/n/12 (\OML/cmm/m/it/12 z; m\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 istheinc
ompleteellipticintegralofthe \OT1/cmr/m/n/12 +
Overfull \hbox (147.59161pt too wide) in paragraph at lines 143--346
\OT1/cmr/m/n/12 +\OML/cmm/m/it/12 firstkind \OT1/cmr/m/n/12 : []\OML/cmm/m/it/1
2 ellipticPi \OT1/cmr/m/n/12 : (\OML/cmm/m/it/12 F; F; F\OT1/cmr/m/n/12 )\OMS/c
msy/m/n/12 ^^@ \OML/cmm/m/it/12 > F \OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 ellipti
cPi\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 z; n; m\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 i
stheincompleteellipticintegralof \OT1/cmr/m/n/12 +
Overfull \hbox (253.87451pt too wide) in paragraph at lines 143--346
\OT1/cmr/m/n/12 +\OML/cmm/m/it/12 thethirdkind \OT1/cmr/m/n/12 : []\OML/cmm/m/i
t/12 jacobiSn \OT1/cmr/m/n/12 : (\OML/cmm/m/it/12 F; F\OT1/cmr/m/n/12 )\OMS/cms
y/m/n/12 ^^@ \OML/cmm/m/it/12 > F \OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 jacobiSn\
OT1/cmr/m/n/12 (\OML/cmm/m/it/12 z; m\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 istheJac
obiellipticsnfunction; defined \OT1/cmr/m/n/12 +
Overfull \hbox (208.28894pt too wide) in paragraph at lines 143--346
\OT1/cmr/m/n/12 +\OML/cmm/m/it/12 bytheformula[]jacobiCn \OT1/cmr/m/n/12 : (\OM
L/cmm/m/it/12 F; F\OT1/cmr/m/n/12 )\OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 > F \O
T1/cmr/m/n/12 + +\OML/cmm/m/it/12 jacobiCn\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 z;
m\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 istheJacobiellipticcnfunction; defined \OT1/
cmr/m/n/12 + +\OML/cmm/m/it/12 by[]and \OT1/cmr/m/n/12 +
Overfull \hbox (91.54376pt too wide) in paragraph at lines 143--346
\OT1/cmr/m/n/12 +[]\OML/cmm/m/it/12 jacobiDn \OT1/cmr/m/n/12 : (\OML/cmm/m/it/1
2 F; F\OT1/cmr/m/n/12 )\OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 > F \OT1/cmr/m/n/1
2 + +\OML/cmm/m/it/12 jacobiDn\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 z; m\OT1/cmr/m/
n/12 )\OML/cmm/m/it/12 istheJacobiellipticdnfunction; defined \OT1/cmr/m/n/12 +
+\OML/cmm/m/it/12 by[]and \OT1/cmr/m/n/12 +
Overfull \hbox (285.62952pt too wide) in paragraph at lines 143--346
\OT1/cmr/m/n/12 +[]\OML/cmm/m/it/12 jacobiZeta \OT1/cmr/m/n/12 : (\OML/cmm/m/it
/12 F; F\OT1/cmr/m/n/12 )\OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 > F \OT1/cmr/m/n
/12 + +\OML/cmm/m/it/12 jacobiZeta\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 z; m\OT1/cm
r/m/n/12 )\OML/cmm/m/it/12 istheJacobiellipticzetafunction; defined \OT1/cmr/m/
n/12 + +\OML/cmm/m/it/12 by[]and \OT1/cmr/m/n/12 +
Overfull \hbox (246.00099pt too wide) in paragraph at lines 143--346
\OT1/cmr/m/n/12 +[]\OML/cmm/m/it/12 :jacobiTheta \OT1/cmr/m/n/12 : (\OML/cmm/m/
it/12 F; F\OT1/cmr/m/n/12 )\OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 > F \OT1/cmr/m
/n/12 + +\OML/cmm/m/it/12 jacobiTheta\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 q; z\OT1
/cmr/m/n/12 )\OML/cmm/m/it/12 isthethirdJacobiThetafunctionlerchPhi \OT1/cmr/m/
n/12 : (\OML/cmm/m/it/12 F; F; F\OT1/cmr/m/n/12 )\OMS/cmsy/m/n/12 ^^@ \OML/cmm/
m/it/12 > F \OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 lerchPhi\OT1/cmr/m/n/12 (\OML/c
mm/m/it/12 z; s; a\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 istheLerchPhifunctionrieman
nZeta \OT1/cmr/m/n/12 :
Overfull \hbox (10.80869pt too wide) in paragraph at lines 143--346
\OML/cmm/m/it/12 F\OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 > F \OT1/cmr/m/n/12 + +
\OML/cmm/m/it/12 riemannZeta\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 z\OT1/cmr/m/n/12
)\OML/cmm/m/it/12 istheRiemannZetafunctioncharlierC \OT1/cmr/m/n/12 : (\OML/cmm
/m/it/12 F; F; F\OT1/cmr/m/n/12 )\OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 > F \OT1
/cmr/m/n/12 + +\OML/cmm/m/it/12 charlierC\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 n; a
; z\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 istheCharlierpolynomialhermiteH \OT1/cmr/m
/n/12 :
Overfull \hbox (296.92938pt too wide) in paragraph at lines 143--346
\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 F; F; F\OT1/cmr/m/n/12 )\OMS/cmsy/m/n/12 ^^@
\OML/cmm/m/it/12 > F \OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 laguerreL\OT1/cmr/m/n/
12 (\OML/cmm/m/it/12 n; a; z\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 istheLaguerrepoly
nomialmeixnerM \OT1/cmr/m/n/12 : (\OML/cmm/m/it/12 F; F; F; F\OT1/cmr/m/n/12 )\
OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 > F \OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 me
ixnerM\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 n; b; c; z\OT1/cmr/m/n/12 )\OML/cmm/m/i
t/12 istheMeixnerpolynomialifFhasRetractableTo\OT1/cmr/m/n/12 (\OML/cmm/m/it/12
Integer\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 thenhypergeometricF \OT1/cmr/m/n/12 :
Overfull \hbox (22.96898pt too wide) in paragraph at lines 143--346
\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 ListF; ListF; F\OT1/cmr/m/n/12 )\OMS/cmsy/m/n
/12 ^^@ \OML/cmm/m/it/12 > F \OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 hypergeometric
F\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 la; lb; z\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 i
sthegeneralizedhypergeometric \OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 functionmeije
rG \OT1/cmr/m/n/12 : (\OML/cmm/m/it/12 ListF; ListF; ListF; ListF; F\OT1/cmr/m/
n/12 )\OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 >
Overfull \hbox (156.48994pt too wide) in paragraph at lines 143--346
\OML/cmm/m/it/12 F \OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 meijerG\OT1/cmr/m/n/12 (
\OML/cmm/m/it/12 la; lb; lc; ld; z\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 isthemeijer
Gfunction \OMS/cmsy/m/n/12 ^^@ ^^@\OML/cmm/m/it/12 Functionsbelowshouldbelocalb
utconditionaliiGamma \OT1/cmr/m/n/12 : \OML/cmm/m/it/12 F\OMS/cmsy/m/n/12 ^^@ \
OML/cmm/m/it/12 > F \OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 iiGamma\OT1/cmr/m/n/12
(\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 shouldbelocalbutconditiona
l\OT1/cmr/m/n/12 ; \OML/cmm/m/it/12 iiabs \OT1/cmr/m/n/12 :
Overfull \hbox (14.12987pt too wide) in paragraph at lines 143--346
\OML/cmm/m/it/12 F \OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 iipolygamma\OT1/cmr/m/n/
12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 shouldbelocalbutconditi
onal\OT1/cmr/m/n/12 ; \OML/cmm/m/it/12 iiBesselJ \OT1/cmr/m/n/12 : \OML/cmm/m/i
t/12 ListF\OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 > F \OT1/cmr/m/n/12 + +\OML/cmm
/m/it/12 iiBesselJ\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 )\OML/cmm/
m/it/12 shouldbelocalbutconditional\OT1/cmr/m/n/12 ; \OML/cmm/m/it/12 iiBesselY
\OT1/cmr/m/n/12 : \OML/cmm/m/it/12 ListF\OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12
>
Overfull \hbox (62.82597pt too wide) in paragraph at lines 143--346
\OML/cmm/m/it/12 F \OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 iAiryBiPrime\OT1/cmr/m/n
/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 shouldbelocalbutcondit
ional\OT1/cmr/m/n/12 ; \OML/cmm/m/it/12 iiHypergeometricF \OT1/cmr/m/n/12 : \OM
L/cmm/m/it/12 ListF\OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 > F \OT1/cmr/m/n/12 +
+\OML/cmm/m/it/12 iiHypergeometricF\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 l\OT1/cmr/
m/n/12 )\OML/cmm/m/it/12 shouldbelocalbutconditional\OT1/cmr/m/n/12 ; \OML/cmm/
m/it/12 iiPolylog \OT1/cmr/m/n/12 :
[2]
Missing $ inserted.
<inserted text>
$
l.406 op_
log_gamma := operator('%logGamma)$CommonOperators
Missing $ inserted.
<inserted text>
$
l.410
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[]\T1/cmr/m/n/12 opabs := operator('abs)$\OML/cmm/m/it/12 CommonOperatorsopconj
ugate \OT1/cmr/m/n/12 := \OML/cmm/m/it/12 operator\OT1/cmr/m/n/12 ([]\OML/cmm/m
/it/12 conjugate\OT1/cmr/m/n/12 )$\T1/cmr/m/n/12 CommonOperators opGamma := ope
rator('Gamma)$\OML/cmm/m/it/12 CommonOperatorsopGamma\OT1/cmr/m/n/12 2 :=
Overfull \hbox (186.6305pt too wide) in paragraph at lines 378--410
\T1/cmr/m/n/12 op-polygamma := operator('polygamma)$\OML/cmm/m/it/12 CommonOper
atorsopBesselJ \OT1/cmr/m/n/12 := \OML/cmm/m/it/12 operator\OT1/cmr/m/n/12 ([]\
OML/cmm/m/it/12 besselJ\OT1/cmr/m/n/12 )$\T1/cmr/m/n/12 CommonOperators opBesse
lY := operator('besselY)$\OML/cmm/m/it/12 CommonOperatorsopBesselI \OT1/cmr/m/n
/12 :=
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\T1/cmr/m/n/12 opAiryAiPrime := operator('airyAiPrime)$\OML/cmm/m/it/12 CommonO
peratorsopAiryBi \OT1/cmr/m/n/12 := \OML/cmm/m/it/12 operator\OT1/cmr/m/n/12 ([
]\OML/cmm/m/it/12 airyBi\OT1/cmr/m/n/12 )$\T1/cmr/m/n/12 CommonOperators opAiry
-BiPrime := operator('airyBiPrime)$\OML/cmm/m/it/12 CommonOperatorsopLambertW \
OT1/cmr/m/n/12 :=
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\OML/cmm/m/it/12 operator\OT1/cmr/m/n/12 ([]\OML/cmm/m/it/12 lambertW\OT1/cmr/m
/n/12 )$\T1/cmr/m/n/12 CommonOperators op-Poly-log := operator('polylog)$\OML/c
mm/m/it/12 CommonOperatorsopWeierstrassP \OT1/cmr/m/n/12 := \OML/cmm/m/it/12 op
erator\OT1/cmr/m/n/12 ([]\OML/cmm/m/it/12 weierstrassP\OT1/cmr/m/n/12 )$\T1/cmr
/m/n/12 CommonOperators
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\T1/cmr/m/n/12 op-Weier-strassP-Prime := operator('weierstrassPPrime)$\OML/cmm/
m/it/12 CommonOperatorsopWeierstrassSigma \OT1/cmr/m/n/12 := \OML/cmm/m/it/12 o
perator\OT1/cmr/m/n/12 ([]\OML/cmm/m/it/12 weierstrassSigma\OT1/cmr/m/n/12 )$\T
1/cmr/m/n/12 CommonOperators
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\T1/cmr/m/n/12 op-Mei-jerG := operator('meijerG)$\OML/cmm/m/it/12 CommonOperato
rsopCharlierC \OT1/cmr/m/n/12 := \OML/cmm/m/it/12 operator\OT1/cmr/m/n/12 ([]\O
ML/cmm/m/it/12 charlierC\OT1/cmr/m/n/12 )$\T1/cmr/m/n/12 CommonOperators opHer-
miteH := operator('hermiteH)$\OML/cmm/m/it/12 CommonOperatorsopJacobiP \OT1/cmr
/m/n/12 :=
Overfull \hbox (63.8742pt too wide) in paragraph at lines 378--410
\OML/cmm/m/it/12 operator\OT1/cmr/m/n/12 ([]\OML/cmm/m/it/12 jacobiP\OT1/cmr/m/
n/12 )$\T1/cmr/m/n/12 CommonOperators opLa-guer-reL := operator('laguerreL)$\OM
L/cmm/m/it/12 CommonOperatorsopMeixnerM \OT1/cmr/m/n/12 := \OML/cmm/m/it/12 ope
rator\OT1/cmr/m/n/12 ([]\OML/cmm/m/it/12 meixnerM\OT1/cmr/m/n/12 )$\T1/cmr/m/n/
12 CommonOperators
[3]
You can't use `macro parameter character #' in horizontal mode.
l.434 nai := #
a
You can't use `macro parameter character #' in horizontal mode.
l.435 nbi := #
b
You can't use `macro parameter character #' in horizontal mode.
l.437 p := (#
a)::F
You can't use `macro parameter character #' in horizontal mode.
l.438 q := (#
b)::F
You can't use `macro parameter character #' in horizontal mode.
l.442 n1 := (#
a)::F
You can't use `macro parameter character #' in horizontal mode.
l.443 n2 := (#
b)::F
You can't use `macro parameter character #' in horizontal mode.
l.444 m1 := (#
c)::F
You can't use `macro parameter character #' in horizontal mode.
l.445 m2 := (#
d)::F
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$
l.452
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$
l.499
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$
l.507
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[]\T1/cmr/m/n/12 elWeierstrassPInverse(l : List F) : F == eWeier-strassPIn-vers
e(l(1), l(2), l(3)) eval-u-ate(opWeierstrassPInverse, elWeierstrassPInverse)$\O
ML/cmm/m/it/12 BasicOperatorFunctions\OT1/cmr/m/n/12 1(\OML/cmm/m/it/12 F\OT1/c
mr/m/n/12 )$
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<inserted text>
$
l.510 eWeierstrassPInverseGrad_
g2(l : List F) : F ==
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<inserted text>
$
l.515
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<inserted text>
$
l.516 eWeierstrassPInverseGrad_
g3(l : List F) : F ==
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<inserted text>
$
l.521
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<inserted text>
$
l.522 eWeierstrassPInverseGrad_
z(l : List F) : F ==
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<inserted text>
$
l.527
Missing $ inserted.
<inserted text>
$
l.528 ...trassPInverse, [eWeierstrassPInverseGrad_
g2,
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<inserted text>
$
l.530
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<inserted text>
$
l.536
[4]
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<inserted text>
$
l.544
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<inserted text>
$
l.545 eWhittakerMGrad_
z(k : F, m : F, z : F) : F ==
Missing $ inserted.
<inserted text>
$
l.548
Missing $ inserted.
<inserted text>
$
l.550 ...rad3(l, t, opWhittakerM, eWhittakerMGrad_
z)
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<inserted text>
$
l.551
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<inserted text>
$
l.558
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<inserted text>
$
l.566
Missing $ inserted.
<inserted text>
$
l.567 eWhittakerWGrad_
z(k : F, m : F, z : F) : F ==
Missing $ inserted.
<inserted text>
$
l.569
Missing $ inserted.
<inserted text>
$
l.571 ...rad3(l, t, opWhittakerW, eWhittakerWGrad_
z)
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<inserted text>
$
l.572
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<inserted text>
$
l.579
Missing $ inserted.
<inserted text>
$
l.590
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<inserted text>
$
l.591 eAngerJGrad_
z(v : F, z : F) : F ==
Missing $ inserted.
<inserted text>
$
l.593
Missing $ inserted.
<inserted text>
$
l.595 ... grad2(l, t, opAngerJ, eAngerJGrad_
z)
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<inserted text>
$
l.596
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<inserted text>
$
l.604
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<inserted text>
$
l.608
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<inserted text>
$
l.614 z = 0 => 2sin(ahalfvpi())^
2/(vpi())
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<inserted text>
$
l.616
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<inserted text>
$
l.619
Missing $ inserted.
<inserted text>
$
l.620 eWeberEGrad_
z(v : F, z : F) : F ==
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<inserted text>
$
l.622
Missing $ inserted.
<inserted text>
$
l.624 ... grad2(l, t, opWeberE, eWeberEGrad_
z)
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<inserted text>
$
l.625
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<inserted text>
$
l.633
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<inserted text>
$
l.637
[5]
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<inserted text>
$
l.645
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<inserted text>
$
l.649 eStruveHGrad_
z(v : F, z : F) : F ==
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<inserted text>
$
l.652
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<inserted text>
$
l.654 ... grad2(l, t, opStruveH, eStruveHGrad_
z)
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<inserted text>
$
l.655
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<inserted text>
$
l.662
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<inserted text>
$
l.670
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<inserted text>
$
l.674 eStruveLGrad_
z(v : F, z : F) : F ==
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<inserted text>
$
l.677
Missing $ inserted.
<inserted text>
$
l.679 ... grad2(l, t, opStruveL, eStruveLGrad_
z)
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<inserted text>
$
l.680
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<inserted text>
$
l.687
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<inserted text>
$
l.695
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<inserted text>
$
l.696 eHankelH1Grad_
z(v : F, z : F) : F ==
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<inserted text>
$
l.698
Missing $ inserted.
<inserted text>
$
l.700 ... grad2(l, t, opHankelH1, eHankelH1Grad_
z)
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<inserted text>
$
l.701
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<inserted text>
$
l.708
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<inserted text>
$
l.716
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<inserted text>
$
l.717 eHankelH2Grad_
z(v : F, z : F) : F ==
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<inserted text>
$
l.719
Missing $ inserted.
<inserted text>
$
l.721 ... grad2(l, t, opHankelH2, eHankelH2Grad_
z)
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<inserted text>
$
l.722
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<inserted text>
$
l.729
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<inserted text>
$
l.737
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<inserted text>
$
l.738 eLommelS1Grad_
z(m : F, v : F, z : F) : F ==
Missing $ inserted.
<inserted text>
$
l.740
[6]
Missing $ inserted.
<inserted text>
$
l.742 ... grad3(l, t, opLommelS1, eLommelS1Grad_
z)
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<inserted text>
$
l.743
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<inserted text>
$
l.750
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<inserted text>
$
l.758
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<inserted text>
$
l.759 eLommelS2Grad_
z(m : F, v : F, z : F) : F ==
Missing $ inserted.
<inserted text>
$
l.761
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<inserted text>
$
l.763 ... grad3(l, t, opLommelS2, eLommelS2Grad_
z)
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<inserted text>
$
l.764
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<inserted text>
$
l.771
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<inserted text>
$
l.780
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<inserted text>
$
l.781 eKummerMGrad_
z(a : F, b : F, z : F) : F ==
Missing $ inserted.
<inserted text>
$
l.783
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<inserted text>
$
l.785 grad3(l, t, opKummerM, eKummerMGrad_
z)
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<inserted text>
$
l.786
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<inserted text>
$
l.793
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<inserted text>
$
l.801
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<inserted text>
$
l.802 eKummerUGrad_
z(a : F, b : F, z : F) : F ==
Missing $ inserted.
<inserted text>
$
l.804
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<inserted text>
$
l.806 grad3(l, t, opKummerU, eKummerUGrad_
z)
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<inserted text>
$
l.807
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<inserted text>
$
l.814
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<inserted text>
$
l.822
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<inserted text>
$
l.823 eLegendrePGrad_
z(nu : F, mu : F, z : F) : F ==
Missing $ inserted.
<inserted text>
$
l.826
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<inserted text>
$
l.828 ... grad3(l, t, opLegendreP, eLegendrePGrad_
z)
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<inserted text>
$
l.829
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<inserted text>
$
l.836
[7]
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<inserted text>
$
l.844
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<inserted text>
$
l.845 eLegendreQGrad_
z(nu : F, mu : F, z : F) : F ==
Missing $ inserted.
<inserted text>
$
l.848
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<inserted text>
$
l.850 ... grad3(l, t, opLegendreQ, eLegendreQGrad_
z)
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<inserted text>
$
l.851
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<inserted text>
$
l.858
Missing $ inserted.
<inserted text>
$
l.866
Missing $ inserted.
<inserted text>
$
l.869 eKelvinBeiGrad_
z(v : F, z : F) : F ==
Missing $ inserted.
<inserted text>
$
l.872
Missing $ inserted.
<inserted text>
$
l.874 ... grad2(l, t, opKelvinBei, eKelvinBeiGrad_
z)
Missing $ inserted.
<inserted text>
$
l.875
Missing $ inserted.
<inserted text>
$
l.882
Missing $ inserted.
<inserted text>
$
l.890
Missing $ inserted.
<inserted text>
$
l.893 eKelvinBerGrad_
z(v : F, z : F) : F ==
Missing $ inserted.
<inserted text>
$
l.896
Missing $ inserted.
<inserted text>
$
l.898 ... grad2(l, t, opKelvinBer, eKelvinBerGrad_
z)
Missing $ inserted.
<inserted text>
$
l.899
Missing $ inserted.
<inserted text>
$
l.906
Missing $ inserted.
<inserted text>
$
l.914
Missing $ inserted.
<inserted text>
$
l.917 eKelvinKeiGrad_
z(v : F, z : F) : F ==
Missing $ inserted.
<inserted text>
$
l.920
Missing $ inserted.
<inserted text>
$
l.922 ... grad2(l, t, opKelvinKei, eKelvinKeiGrad_
z)
Missing $ inserted.
<inserted text>
$
l.923
Missing $ inserted.
<inserted text>
$
l.930
Missing $ inserted.
<inserted text>
$
l.938
Missing $ inserted.
<inserted text>
$
l.941 eKelvinKerGrad_
z(v : F, z : F) : F ==
Missing $ inserted.
<inserted text>
$
l.944
[8]
Missing $ inserted.
<inserted text>
$
l.946 ... grad2(l, t, opKelvinKer, eKelvinKerGrad_
z)
Missing $ inserted.
<inserted text>
$
l.947
Missing $ inserted.
<inserted text>
$
l.954
Missing $ inserted.
<inserted text>
$
l.962
Missing $ inserted.
<inserted text>
$
l.971
Missing $ inserted.
<inserted text>
$
l.979
Missing $ inserted.
<inserted text>
$
l.988
Missing $ inserted.
<inserted text>
$
l.998
Missing $ inserted.
<inserted text>
$
l.1001 eEllipticE2Grad_
z(l : List F) : F ==
Missing $ inserted.
<inserted text>
$
l.1005
Missing $ inserted.
<inserted text>
$
l.1006 eEllipticE2Grad_
m(l : List F) : F ==
Missing $ inserted.
<inserted text>
$
l.1010
Missing $ inserted.
<inserted text>
$
l.1011 ...rivative(opEllipticE2, [eEllipticE2Grad_
z, eEllipticE2Grad_m])
Missing $ inserted.
<inserted text>
$
l.1012
Missing $ inserted.
<inserted text>
$
l.1021
Missing $ inserted.
<inserted text>
$
l.1031
Missing $ inserted.
<inserted text>
$
l.1034 eEllipticFGrad_
z(l : List F) : F ==
Missing $ inserted.
<inserted text>
$
l.1038
Missing $ inserted.
<inserted text>
$
l.1039 eEllipticFGrad_
m(l : List F) : F ==
Missing $ inserted.
<inserted text>
$
l.1044
Missing $ inserted.
<inserted text>
$
l.1045 ...derivative(opEllipticF, [eEllipticFGrad_
z, eEllipticFGrad_m])
Missing $ inserted.
<inserted text>
$
l.1046
[9]
Missing $ inserted.
<inserted text>
$
l.1050
Missing $ inserted.
<inserted text>
$
l.1059
Missing $ inserted.
<inserted text>
$
l.1062 eEllipticPiGrad_
z(l : List F) : F ==
Missing $ inserted.
<inserted text>
$
l.1067
Missing $ inserted.
<inserted text>
$
l.1068 eEllipticPiGrad_
n(l : List F) : F ==
Missing $ inserted.
<inserted text>
$
l.1078
Missing $ inserted.
<inserted text>
$
l.1079 eEllipticPiGrad_
m(l : List F) : F ==
Missing $ inserted.
<inserted text>
$
l.1086
Missing $ inserted.
<inserted text>
$
l.1087 ...rivative(opEllipticPi, [eEllipticPiGrad_
z, eEllipticPiGrad_n,
Missing $ inserted.
<inserted text>
$
l.1089
Missing $ inserted.
<inserted text>
$
l.1093
Missing $ inserted.
<inserted text>
$
l.1106
Missing $ inserted.
<inserted text>
$
l.1110 eJacobiSnGrad_
z(l : List F) : F ==
Missing $ inserted.
<inserted text>
$
l.1114
Missing $ inserted.
<inserted text>
$
l.1115 eJacobiSnGrad_
m(l : List F) : F ==
Missing $ inserted.
<inserted text>
$
l.1120
Missing $ inserted.
<inserted text>
$
l.1121 derivative(opJacobiSn, [eJacobiSnGrad_
z, eJacobiSnGrad_m])
Missing $ inserted.
<inserted text>
$
l.1122
Missing $ inserted.
<inserted text>
$
l.1126
Missing $ inserted.
<inserted text>
$
l.1135
Missing $ inserted.
<inserted text>
$
l.1136 eJacobiCnGrad_
z(l : List F) : F ==
Missing $ inserted.
<inserted text>
$
l.1140
Missing $ inserted.
<inserted text>
$
l.1141 eJacobiCnGrad_
m(l : List F) : F ==
Missing $ inserted.
<inserted text>
$
l.1146
Missing $ inserted.
<inserted text>
$
l.1147 derivative(opJacobiCn, [eJacobiCnGrad_
z, eJacobiCnGrad_m])
Missing $ inserted.
<inserted text>
$
l.1148
Missing $ inserted.
<inserted text>
$
l.1152
Missing $ inserted.
<inserted text>
$
l.1161
Missing $ inserted.
<inserted text>
$
l.1162 eJacobiDnGrad_
z(l : List F) : F ==
Missing $ inserted.
<inserted text>
$
l.1166
[10]
Missing $ inserted.
<inserted text>
$
l.1167 eJacobiDnGrad_
m(l : List F) : F ==
Missing $ inserted.
<inserted text>
$
l.1172
Missing $ inserted.
<inserted text>
$
l.1173 derivative(opJacobiDn, [eJacobiDnGrad_
z, eJacobiDnGrad_m])
Missing $ inserted.
<inserted text>
$
l.1174
Missing $ inserted.
<inserted text>
$
l.1178
Missing $ inserted.
<inserted text>
$
l.1187
Missing $ inserted.
<inserted text>
$
l.1188 eJacobiZetaGrad_
z(l : List F) : F ==
Missing $ inserted.
<inserted text>
$
l.1193
Missing $ inserted.
<inserted text>
$
l.1194 eJacobiZetaGrad_
m(l : List F) : F ==
Missing $ inserted.
<inserted text>
$
l.1206
Missing $ inserted.
<inserted text>
$
l.1207 ...rivative(opJacobiZeta, [eJacobiZetaGrad_
z, eJacobiZetaGrad_m])
Missing $ inserted.
<inserted text>
$
l.1208
Missing $ inserted.
<inserted text>
$
l.1212
Missing $ inserted.
<inserted text>
$
l.1220
Missing $ inserted.
<inserted text>
$
l.1224
Missing $ inserted.
<inserted text>
$
l.1228 -- z = 0 => 1/a^
s
Missing $ inserted.
<inserted text>
$
l.1231
Missing $ inserted.
<inserted text>
$
l.1234
Missing $ inserted.
<inserted text>
$
l.1252
Missing $ inserted.
<inserted text>
$
l.1260
Missing $ inserted.
<inserted text>
$
l.1274
[11]
Missing $ inserted.
<inserted text>
$
l.1287
Missing $ inserted.
<inserted text>
$
l.1288 eHermiteHGrad_
z(n : F, z : F) : F == (2::F)nhermiteH(n - 1, z)
Missing $ inserted.
<inserted text>
$
l.1289
Missing $ inserted.
<inserted text>
$
l.1291 ... grad2(l, t, opHermiteH, eHermiteHGrad_
z)
Missing $ inserted.
<inserted text>
$
l.1292
Missing $ inserted.
<inserted text>
$
l.1308
Missing $ inserted.
<inserted text>
$
l.1309 eJacobiPGrad_
z(n : F, a : F, b : F, z : F) : F ==
Missing $ inserted.
<inserted text>
$
l.1311
Missing $ inserted.
<inserted text>
$
l.1313 ... grad4(l, t, opJacobiP, eJacobiPGrad_
z)
Missing $ inserted.
<inserted text>
$
l.1314
Missing $ inserted.
<inserted text>
$
l.1330
Missing $ inserted.
<inserted text>
$
l.1331 eLaguerreLGrad_
z(n : F, a : F, z : F) : F ==
Missing $ inserted.
<inserted text>
$
l.1333
Missing $ inserted.
<inserted text>
$
l.1335 ...grad3(l, t, opLaguerreL, eLaguerreLGrad_
z)
Missing $ inserted.
<inserted text>
$
l.1336
Missing $ inserted.
<inserted text>
$
l.1351
[12]
Overfull \hbox (34.30756pt too wide) in paragraph at lines 1437--1451
[]\T1/cmr/m/n/12 iconjugate(k:K):F == --output("in: ",k::OutputForm)$\OML/cmm/m
/it/12 OutputPackageifis\OT1/cmr/m/n/12 ?(\OML/cmm/m/it/12 k; opconjugate\OT1/c
mr/m/n/12 )\OML/cmm/m/it/12 thenx \OT1/cmr/m/n/12 := \OML/cmm/m/it/12 argument\
OT1/cmr/m/n/12 (\OML/cmm/m/it/12 k\OT1/cmr/m/n/12 )(1)\OML/cmm/m/it/12 elseifis
\OT1/cmr/m/n/12 ?(\OML/cmm/m/it/12 k; opabs\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 th
enx \OT1/cmr/m/n/12 :=
Overfull \hbox (140.67513pt too wide) in paragraph at lines 1437--1451
\OML/cmm/m/it/12 k \OT1/cmr/m/n/12 :: \OML/cmm/m/it/12 FelseifsymbolIfCan\OT1/c
mr/m/n/12 (\OML/cmm/m/it/12 k\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 caseSymbolthenx
\OT1/cmr/m/n/12 := \OML/cmm/m/it/12 kernel\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 opc
onjugate; k \OT1/cmr/m/n/12 :: \OML/cmm/m/it/12 F\OT1/cmr/m/n/12 )\OML/cmm/m/it
/12 elseifFhasRadicalCategoryandRhasRetractableTo\OT1/cmr/m/n/12 (\OML/cmm/m/it
/12 Integer\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 andis\OT1/cmr/m/n/12 ?(\OML/cmm/m/
it/12 k;[] nthRoot\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 thenx \OT1/cmr/m/n/12 :=
Overfull \hbox (34.8454pt too wide) in paragraph at lines 1437--1451
\OT1/cmr/m/n/12 1\OML/cmm/m/it/12 =nthRoot\OT1/cmr/m/n/12 (1\OML/cmm/m/it/12 =i
iconjugateargument\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 k\OT1/cmr/m/n/12 )(1)\OML/c
mm/m/it/12 ; retract\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 argument\OT1/cmr/m/n/12 (
\OML/cmm/m/it/12 k\OT1/cmr/m/n/12 )(2))@\OML/cmm/m/it/12 Integer\OT1/cmr/m/n/12
)\OML/cmm/m/it/12 else \OMS/cmsy/m/n/12 ^^@ ^^@\OML/cmm/m/it/12 assumeholomorp
hicx \OT1/cmr/m/n/12 := \OML/cmm/m/it/12 map\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 i
iconjugate; k\OT1/cmr/m/n/12 ) \OMS/cmsy/m/n/12 ^^@ ^^@\OML/cmm/m/it/12 output\
OT1/cmr/m/n/12 ("\OML/cmm/m/it/12 out \OT1/cmr/m/n/12 :
Missing $ inserted.
<inserted text>
$
l.1456 x := reduce(_
+$F,map(iiconjugate,s))
You can't use `macro parameter character #' in horizontal mode.
l.1460 else if #
(ks:List K:=kernels(x))>0 then
Missing $ inserted.
<inserted text>
$
l.1470
Overfull \hbox (100.31317pt too wide) in paragraph at lines 1452--1470
\OML/cmm/m/it/12 OutputForm\OT1/cmr/m/n/12 )$\T1/cmr/m/n/12 OutputPackage x :=
reduce($[]$F,map(iiconjugate,s)) else if (s:=isTimes(x)) case List F then --out
put("isTimes: ",s::OutputForm)$\OML/cmm/m/it/12 OutputPackagex \OT1/cmr/m/n/12
:=
Overfull \hbox (20.36736pt too wide) in paragraph at lines 1452--1470
\OML/cmm/m/it/12 reduce\OT1/cmr/m/n/12 ([]$\T1/cmr/m/n/12 F,map(iiconjugate,s))
else if (ks:List K:=kernels(x))>0 then --output("kernels: ",ks::OutputForm)$\O
ML/cmm/m/it/12 OutputPackagex \OT1/cmr/m/n/12 := \OML/cmm/m/it/12 eval\OT1/cmr/
m/n/12 (\OML/cmm/m/it/12 x; ks;[] ap\OT1/cmr/m/n/12 ((\OML/cmm/m/it/12 k \OT1/c
mr/m/n/12 :
Missing $ inserted.
<inserted text>
$
l.1483 zero?(x) => 1::F/((3::F)^
twothirds*Gamma(twothirds))
Missing $ inserted.
<inserted text>
$
l.1499
Overfull \hbox (79.65605pt too wide) in paragraph at lines 1481--1499
[]\T1/cmr/m/n/12 if F has El-e-men-tary-Func-tion-Cat-e-gory then iAiryAi x ==
zero?(x) => 1::F/((3::F)$[]\OML/cmm/m/it/12 wothirds \OMS/cmsy/m/n/12 ^^C \OML/
cmm/m/it/12 Gamma\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 twothirds\OT1/cmr/m/n/12 ))\
OML/cmm/m/it/12 kernel\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 opAiryAi; x\OT1/cmr/m/n
/12 )\OML/cmm/m/it/12 iAiryAiPrimex \OT1/cmr/m/n/12 ==
Overfull \hbox (54.76915pt too wide) in paragraph at lines 1481--1499
\OML/cmm/m/it/12 Gamma\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 twothirds\OT1/cmr/m/n/1
2 ))\OML/cmm/m/it/12 kernel\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 opAiryBi; x\OT1/cm
r/m/n/12 )\OML/cmm/m/it/12 iAiryBiPrimex \OT1/cmr/m/n/12 == \OML/cmm/m/it/12 ze
ro\OT1/cmr/m/n/12 ?