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last edited 9 years ago by Bill Page |
Edit detail for SandBoxFunctionalSpecialFunction revision 9 of 33
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Editor: Bill Page
Time: 2014/08/13 23:20:41 GMT+0 |
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| Note: conjugate | ||
removed:
- --(a := retractIfCan(numer f)@Union(R, "failed")) case "failed" or
- -- (b := retractIfCan(denom f)@Union(R,"failed")) case "failed" => iconjugate x
- --conjugate(a::R)::F / conjugate(b::R)::F
Changed derivative of abs(x) to
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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, %i*z)} 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-m*t^2)), t = 0..1)} ellipticE : F -> F ++ ellipticE(m) is the complete elliptic integral of the ++ second kind: \spad{ellipticE(m) = ++ integrate(sqrt(1-m*t^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-m*t^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-m*t^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-n*t^2)*sqrt((1-t^2)*(1-m*t^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 iconjugate: 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 := 2*athird threehalfs : F := 3*ahalf
-- 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(4*z^3 - g2*z - 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)$CommonOperators
angerJ(v,z) == opAngerJ(v, z)
if F has TranscendentalFunctionCategory then
eAngerJ(v : F,z : F) : F == z = 0 => sin(v*pi())/(v*pi()) kernel(opAngerJ, [v, z])
elAngerJ(l : List F) : F == eAngerJ(l(1),l(2)) evaluate(opAngerJ, elAngerJ)$BasicOperatorFunctions1(F)
eAngerJGrad_z(v : F,z : F) : F == -angerJ(v + 1, z) + v*angerJ(v, z)/z - sin(v*pi())/(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 => 2*sin(ahalf*v*pi())^2/(v*pi()) kernel(opWeberE, [v, z])
elWeberE(l : List F) : F == eWeberE(l(1),l(2)) evaluate(opWeberE, elWeberE)$BasicOperatorFunctions1(F)
eWeberEGrad_z(v : F,z : F) : F == -weberE(v + 1, z) + v*weberE(v, z)/z - (1 - cos(v*pi()))/(pi()*z)
dWeberE(l : List F,t : SE) : F == grad2(l, t, opWeberE, eWeberEGrad_z)
setProperty(opWeberE,SPECIALDIFF, dWeberE@((List F, SE)->F) pretend None)
else
eeWeberE(l : List F) : F == kernel(opWeberE,l) evaluate(opWeberE, eeWeberE)$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) + v*struveH(v, z)/z + (ahalf*z)^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) + v*struveL(v, z)/z + (ahalf*z)^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 == -v*lommelS1(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 == -v*lommelS2(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 == ahalf*sqrt(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 == ahalf*sqrt(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 == ahalf*sqrt(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 == ahalf*sqrt(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 - m*z^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 - z*sqrt(1 - z^2)/sqrt(1 - m*z^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 - n*z^2)*sqrt(1 - m*z^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 := n*z*sqrt(1 - m*z^2)*sqrt(1 - z^2)/ ((1 - n*z^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 := m*z*sqrt(1 - z^2)/sqrt(1 - m*z^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) -m*jacobiSn(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 := (dn*dn + m - 1)*jacobiZeta(z, m) res2 := res1 + (m - 1)*z*dn*dn res3 := res2 - m*jacobiCn(z, m)*jacobiDn(z, m)*jacobiSn(z, m) res4 := res3 + z*(1 - m + dn*dn)*er ahalf*(res4 - z*er*er)/(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) - a*lerchPhi(z, s, a))/z da := -differentiate(a, t)*s*lerchPhi(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)*n*hermiteH(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/(c*b) + 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 is?(x,opabs) => x smaller?(x, 0) => kernel(opabs, -x) kernel(opabs, x)
iconjugate z == zero? z => 0 is?(z,opconjugate) => argument(retract(z)@K)(1) kernel(opconjugate, z)
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)^twothirds*Gamma(twothirds)) kernel(opAiryAi,x) iAiryAiPrime x == zero?(x) => -1::F/((3::F)^athird*Gamma(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$F kernel(opLambertW,x) else iLambertW(x) == zero?(x) => 0 kernel(opLambertW, x)
if F has ElementaryFunctionCategory then if F has LiouvillianFunctionCategory then iiPolylog(s,x) == s = 1 => -log(1 - x) s = 2::F => dilog(1 - x) kernel(opPolylog, [s, x]) else iiPolylog(s, x) == s = 1 => -log(1 - x) kernel(opPolylog, [s, x]) else iiPolylog(s, x) == kernel(opPolylog, [s, x])
iPolylog(l) == iiPolylog(first l,second l)
iWeierstrassP(g2,g3, x) == kernel(opWeierstrassP, [g2, g3, x]) iWeierstrassPPrime(g2, g3, x) == kernel(opWeierstrassPPrime, [g2, g3, x])
iWeierstrassSigma(g2,g3, x) == x = 0 => 0 kernel(opWeierstrassSigma, [g2, g3, x])
iWeierstrassZeta(g2,g3, x) == kernel(opWeierstrassZeta, [g2, g3, x])
-- Could put more conditional special rules for other functions here
if R has abs : R -> R then import from Polynomial R iiabs x == (r := retractIfCan(x)@Union(Fraction Polynomial R,"failed")) case "failed" => iabs x f := r::Fraction Polynomial R (a := retractIfCan(numer f)@Union(R, "failed")) case "failed" or (b := retractIfCan(denom f)@Union(R, "failed")) case "failed" => iabs x abs(a::R)::F / abs(b::R)::F else iiabs x == iabs x
if R has conjugate : R -> R then import from Polynomial R iiconjugate x == (r := retractIfCan(x)@Union(Fraction Polynomial R,"failed")) case "failed" => iconjugate x f := r::Fraction Polynomial R map(conjugate$R, numer f)$PolynomialFunctions2(R, R)::F / _ map(conjugate$R, denom f)$PolynomialFunctions2(R, R)::F else iiconjugate x == 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 - 27*g3^2 wp := weierstrassP(g2,g3, x) (weierstrassPPrime(g2, g3, x)*(-9*ahalf*g3 *weierstrassZeta(g2, g3, x) + afourth*g2^2*x) - 9*g3*wp^2 + ahalf*g2^2*wp + 3*ahalf*g2*g3)/delta
iWeierstrassPGrad2(l : List F) : F == g2 := first l g3 := second l x := third l delta := g2^3 - 27*g3^2 wp := weierstrassP(g2,g3, x) (weierstrassPPrime(g2, g3, x)*(3*g2*weierstrassZeta(g2, g3, x) - 9*ahalf*g3*x) + 6*g2*wp^2 - 9*g3*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 - 27*g3^2 wp := weierstrassP(g2,g3, x) wpp := weierstrassPPrime(g2, g3, x) wpp2 := 6*wp^2 - ahalf*g2 (wpp2*(-9*ahalf*g3*weierstrassZeta(g2, g3, x) + afourth*g2^2*x) + wpp*(9*ahalf*g3*wp + afourth*g2^2) - 18*g3*wp*wpp + ahalf*g2^2*wpp)/delta
iWeierstrassPPrimeGrad2(l : List F) : F == g2 := first l g3 := second l x := third l delta := g2^3 - 27*g3^2 wp := weierstrassP(g2,g3, x) wpp := weierstrassPPrime(g2, g3, x) wpp2 := 6*wp^2 - ahalf*g2 (wpp2*(3*g2*weierstrassZeta(g2, g3, x) - 9*ahalf*g3*x) + wpp*(-3*g2*wp - 9*ahalf*g3) + 12*g2*wp*wpp - 9*g3*wpp)/delta
iWeierstrassPPrimeGrad3(l : List F) : F == g2 := first l 6*weierstrassP(g2,second l, third l)^2 - ahalf*g2
iWeierstrassSigmaGrad1(l : List F) : F == g2 := first l g3 := second l x := third l delta := g2^3 - 27*g3^2 ws := weierstrassSigma(g2,g3, x) wz := weierstrassZeta(g2, g3, x) wsp := wz*ws wsp2 := - weierstrassP(g2, g3, x)*ws + wz^2*ws afourth*(-9*g3*wsp2 - g2^2*ws - 3*afourth*g2*g3*x^2*ws + g2^2*x*wsp)/delta
iWeierstrassSigmaGrad2(l : List F) : F == g2 := first l g3 := second l x := third l delta := g2^3 - 27*g3^2 ws := weierstrassSigma(g2,g3, x) wz := weierstrassZeta(g2, g3, x) wsp := wz*ws wsp2 := - weierstrassP(g2, g3, x)*ws + wz^2*ws ahalf*(3*g2*wsp2 + 9*g3*ws + afourth*g2^2*x^2*ws - 9*g3*x*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 - 27*g3^2 wp := weierstrassP(g2,g3, x) (ahalf*weierstrassZeta(g2, g3, x)*(9*g3*wp + ahalf*g2^2) - ahalf*g2*x*(ahalf*g2*wp+3*afourth*g3) + 9*afourth*g3*weierstrassPPrime(g2, g3, x))/delta
iWeierstrassZetaGrad2(l : List F) : F == g2 := first l g3 := second l x := third l delta := g2^3 - 27*g3^2 wp := weierstrassP(g2,g3, x) (-3*weierstrassZeta(g2, g3, x)*(g2*wp + 3*ahalf*g3) + ahalf*x*(9*g3*wp + ahalf*g2^2) - 3*ahalf*g2*weierstrassPPrime(g2, g3, x))/delta
iWeierstrassZetaGrad3(l : List F) : F == -weierstrassP(first l,second l, third l)
OF ==> OutputForm SEX ==> SExpression NNI ==> NonNegativeInteger
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)*aprod*opHypergeometricF(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)*(sign*nm + 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)$SEX rl := cons(p, rl) lo := rest(lo) rl := reverse!(rl) rl
dhyper0(op : OF,lo : List OF) : OF == n0 := (#lo - 2)::NNI pql := get_int_listo(2, lo) lo := first(lo, n0) al := get_of_lists(pql, lo) lo := first(al) al := rest(al) a := first al b := first(rest(al)) z := first(lo) prefix(op, [bracket a, bracket b, z])
dhyper(lo : List OF) : OF == dhyper0("hypergeometricF"::Symbol::OF,lo)
ddhyper(lo : List OF) : OF == dhyper0(first lo,rest lo)
dmeijer0(op : OF,lo : List OF) : OF == n0 := (#lo - 4)::NNI nl := get_int_listo(4, lo) lo := first(lo, n0) al := get_of_lists(nl, lo) lo := first(al) al := rest(al) z := first(lo) prefix(op, concat( map(bracket, al)$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 == 2*x*kernel(op_erfs,x) - 2::F/sqrt(pi())
d_erfis(x : F) : F == -2*x*kernel(op_erfis,x) + 2::F/sqrt(pi())
derivative(op_erfs,d_erfs) derivative(op_erfis, d_erfis)
derivative(opabs,(x : F) : F +-> x*inv(abs(x))) derivative(opconjugate, (x : F) : F +-> conjugate(x)) 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 +-> x*airyAi(x)) derivative(opAiryBi, (x : F) : F +-> airyBiPrime(x)) derivative(opAiryBiPrime, (x : F) : F +-> x*airyBi(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/9176836072375115294-25px001.spad
using old system compiler.
FSPECX abbreviates package FunctionalSpecialFunction
(EVAL-WHEN (EVAL LOAD) (SETQ |$tryRecompileArguments| NIL))
Value = NIL
------------------------------------------------------------------------
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: 0.06 SEC.
compiling exported conjugate : F -> F
Time: 0 SEC.
compiling exported Gamma : F -> F
Time: 0 SEC.
compiling exported Gamma : (F,
compiling exported Beta : (F,
compiling exported digamma : F -> F
Time: 0 SEC.
compiling exported polygamma : (F,
compiling exported besselJ : (F,
compiling exported besselY : (F,
compiling exported besselI : (F,
compiling exported besselK : (F,
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,
compiling exported weierstrassP : (F,
compiling exported weierstrassPPrime : (F,
compiling exported weierstrassSigma : (F,
compiling exported weierstrassZeta : (F,
****** Domain: F already in scope
augmenting F: (RetractableTo (Integer))
augmenting $: (SIGNATURE $ hypergeometricF (F (List F) (List F) F))
augmenting $: (SIGNATURE $ meijerG (F (List F) (List F) (List F) (List F) F))
compiling exported hypergeometricF : (List F,
compiling exported meijerG : (List F,
importing List Kernel F
processing macro definition dummy ==> ::((Sel (Symbol) new),
compiling local grad3 : (List F,
compiling local grad4 : (List F,
compiling exported whittakerM : (F,
compiling local eWhittakerM : (F,
compiling local elWhittakerM : List F -> F
Time: 0 SEC.
compiling local eWhittakerMGrad_z : (F,
compiling local dWhittakerM : (List F,
compiling exported whittakerW : (F,
compiling local eWhittakerW : (F,
compiling local elWhittakerW : List F -> F
Time: 0 SEC.
compiling local eWhittakerWGrad_z : (F,
compiling local dWhittakerW : (List F,
compiling exported angerJ : (F,
****** Domain: F already in scope
augmenting F: (TranscendentalFunctionCategory)
compiling local eAngerJ : (F,
compiling local elAngerJ : List F -> F
Time: 0 SEC.
compiling local eAngerJGrad_z : (F,
compiling local dAngerJ : (List F,
compiling local eeAngerJ : List F -> F
Time: 0 SEC.
compiling exported weberE : (F,
****** Domain: F already in scope
augmenting F: (TranscendentalFunctionCategory)
compiling local eWeberE : (F,
compiling local elWeberE : List F -> F
Time: 0.01 SEC.
compiling local eWeberEGrad_z : (F,
compiling local dWeberE : (List F,
compiling local eeWeberE : List F -> F
Time: 0.01 SEC.
compiling exported struveH : (F,
compiling local eStruveH : (F,
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,
compiling local dStruveH : (List F,
compiling exported struveL : (F,
compiling local eStruveL : (F,
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,
compiling local dStruveL : (List F,
compiling exported hankelH1 : (F,
compiling local eHankelH1 : (F,
compiling local elHankelH1 : List F -> F
Time: 0.01 SEC.
compiling local eHankelH1Grad_z : (F,
compiling local dHankelH1 : (List F,
compiling exported hankelH2 : (F,
compiling local eHankelH2 : (F,
compiling local elHankelH2 : List F -> F
Time: 0 SEC.
compiling local eHankelH2Grad_z : (F,
compiling local dHankelH2 : (List F,
compiling exported lommelS1 : (F,
compiling local eLommelS1 : (F,
compiling local elLommelS1 : List F -> F
Time: 0 SEC.
compiling local eLommelS1Grad_z : (F,
compiling local dLommelS1 : (List F,
compiling exported lommelS2 : (F,
compiling local eLommelS2 : (F,
compiling local elLommelS2 : List F -> F
Time: 0 SEC.
compiling local eLommelS2Grad_z : (F,
compiling local dLommelS2 : (List F,
compiling exported kummerM : (F,
compiling local eKummerM : (F,
compiling local elKummerM : List F -> F
Time: 0 SEC.
compiling local eKummerMGrad_z : (F,
compiling local dKummerM : (List F,
compiling exported kummerU : (F,
compiling local eKummerU : (F,
compiling local elKummerU : List F -> F
Time: 0 SEC.
compiling local eKummerUGrad_z : (F,
compiling local dKummerU : (List F,
compiling exported legendreP : (F,
compiling local eLegendreP : (F,
compiling local elLegendreP : List F -> F
Time: 0 SEC.
compiling local eLegendrePGrad_z : (F,
compiling local dLegendreP : (List F,
compiling exported legendreQ : (F,
compiling local eLegendreQ : (F,
compiling local elLegendreQ : List F -> F
Time: 0 SEC.
compiling local eLegendreQGrad_z : (F,
compiling local dLegendreQ : (List F,
compiling exported kelvinBei : (F,
compiling local eKelvinBei : (F,
compiling local elKelvinBei : List F -> F
Time: 0 SEC.
****** Domain: F already in scope
augmenting F: (RadicalCategory)
compiling local eKelvinBeiGrad_z : (F,
compiling local dKelvinBei : (List F,
compiling exported kelvinBer : (F,
compiling local eKelvinBer : (F,
compiling local elKelvinBer : List F -> F
Time: 0 SEC.
****** Domain: F already in scope
augmenting F: (RadicalCategory)
compiling local eKelvinBerGrad_z : (F,
compiling local dKelvinBer : (List F,
compiling exported kelvinKei : (F,
compiling local eKelvinKei : (F,
compiling local elKelvinKei : List F -> F
Time: 0 SEC.
****** Domain: F already in scope
augmenting F: (RadicalCategory)
compiling local eKelvinKeiGrad_z : (F,
compiling local dKelvinKei : (List F,
compiling exported kelvinKer : (F,
compiling local eKelvinKer : (F,
compiling local elKelvinKer : List F -> F
Time: 0 SEC.
****** Domain: F already in scope
augmenting F: (RadicalCategory)
compiling local eKelvinKerGrad_z : (F,
compiling local dKelvinKer : (List F,
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: 0.01 SEC.
compiling exported ellipticE : F -> F
Time: 0.01 SEC.
compiling local eEllipticE : F -> F
Time: 0 SEC.
compiling local elEllipticE : List F -> F
Time: 0 SEC.
compiling local dEllipticE : F -> F
Time: 0.01 SEC.
compiling exported ellipticE : (F,
compiling local eEllipticE2 : (F,
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: 0.01 SEC.
compiling local inEllipticE2 : List InputForm -> InputForm
Time: 0.01 SEC.
compiling exported ellipticF : (F,
compiling local eEllipticF : (F,
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: 0.01 SEC.
compiling local eEllipticFGrad_m : List F -> F
Time: 0.02 SEC.
compiling exported ellipticPi : (F,
compiling local eEllipticPi : (F,
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: 0.13 SEC.
compiling local eEllipticPiGrad_n : List F -> F
Time: 0.22 SEC.
compiling local eEllipticPiGrad_m : List F -> F
Time: 0.02 SEC.
compiling exported jacobiSn : (F,
compiling local eJacobiSn : (F,
compiling local elJacobiSn : List F -> F
Time: 0 SEC.
compiling local jacobiGradHelper : (F,
compiling local eJacobiSnGrad_z : List F -> F
Time: 0.01 SEC.
compiling local eJacobiSnGrad_m : List F -> F
Time: 0.01 SEC.
compiling exported jacobiCn : (F,
compiling local eJacobiCn : (F,
compiling local elJacobiCn : List F -> F
Time: 0.01 SEC.
compiling local eJacobiCnGrad_z : List F -> F
Time: 0 SEC.
compiling local eJacobiCnGrad_m : List F -> F
Time: 0.03 SEC.
compiling exported jacobiDn : (F,
compiling local eJacobiDn : (F,
compiling local elJacobiDn : List F -> F
Time: 0 SEC.
compiling local eJacobiDnGrad_z : List F -> F
Time: 0.01 SEC.
compiling local eJacobiDnGrad_m : List F -> F
Time: 0.04 SEC.
compiling exported jacobiZeta : (F,
compiling local eJacobiZeta : (F,
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: 0.26 SEC.
compiling exported jacobiTheta : (F,
compiling local eJacobiTheta : (F,
compiling local elJacobiTheta : List F -> F
Time: 0 SEC.
compiling exported lerchPhi : (F,
compiling local eLerchPhi : (F,
compiling local elLerchPhi : List F -> F
Time: 0.01 SEC.
compiling local dLerchPhi : (List F,
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,
compiling local eCharlierC : (F,
compiling local elCharlierC : List F -> F
Time: 0.01 SEC.
compiling exported hermiteH : (F,
compiling local eHermiteH : (F,
compiling local elHermiteH : List F -> F
Time: 0 SEC.
compiling local eHermiteHGrad_z : (F,
compiling local dHermiteH : (List F,
compiling exported jacobiP : (F,
compiling local eJacobiP : (F,
compiling local elJacobiP : List F -> F
Time: 0 SEC.
compiling local eJacobiPGrad_z : (F,
compiling local dJacobiP : (List F,
compiling exported laguerreL : (F,
compiling local eLaguerreL : (F,
compiling local elLaguerreL : List F -> F
Time: 0 SEC.
compiling local eLaguerreLGrad_z : (F,
compiling local dLaguerreL : (List F,
compiling exported meixnerM : (F,
compiling local eMeixnerM : (F,
compiling local elMeixnerM : List F -> F
Time: 0 SEC.
compiling exported belong? : BasicOperator -> Boolean
Time: 0 SEC.
compiling exported operator : BasicOperator -> BasicOperator
Time: 0.03 SEC.
compiling local iGamma : F -> F
Time: 0.01 SEC.
compiling local iabs : F -> F
Time: 0 SEC.
compiling local iconjugate : F -> F
Time: 0 SEC.
compiling local iBeta : (F,
compiling local idigamma : F -> F
Time: 0 SEC.
compiling local iiipolygamma : (F,
compiling local iiiBesselJ : (F,
compiling local iiiBesselY : (F,
compiling local iiiBesselI : (F,
compiling local iiiBesselK : (F,
importing Fraction Integer
****** Domain: F already in scope
augmenting F: (ElementaryFunctionCategory)
compiling exported iAiryAi : F -> F
Time: 0.01 SEC.
compiling exported iAiryAiPrime : F -> F
Time: 0.01 SEC.
compiling exported iAiryBi : F -> F
Time: 0.01 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: 0.01 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,
compiling exported iiPolylog : (F,
compiling exported iiPolylog : (F,
compiling local iPolylog : List F -> F
Time: 0 SEC.
compiling local iWeierstrassP : (F,
compiling local iWeierstrassPPrime : (F,
compiling local iWeierstrassSigma : (F,
compiling local iWeierstrassZeta : (F,
augmenting R: (SIGNATURE R abs (R R))
importing Polynomial R
compiling exported iiabs : F -> F
Time: 0.02 SEC.
compiling exported iiabs : F -> F
Time: 0 SEC.
augmenting R: (SIGNATURE R conjugate (R R))
importing Polynomial R
compiling exported iiconjugate : F -> F
Time: 0.01 SEC.
compiling exported iiconjugate : F -> F
FSPECX;iiconjugate;2F;229 is replaced by x
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: 0.01 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: 0.01 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: 0.01 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,
compiling local iBesselJ : (List F,
compiling local iBesselY : (List F,
compiling local iBesselI : (List F,
compiling local iBesselK : (List F,
compiling local dPolylog : (List F,
compiling local ipolygamma : (List F,
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,
compiling local inGamma2 : List InputForm -> InputForm
Time: 0.01 SEC.
compiling local dLambertW : F -> F
Time: 0 SEC.
compiling local iWeierstrassPGrad1 : List F -> F
Time: 0.42 SEC.
compiling local iWeierstrassPGrad2 : List F -> F
Time: 0.14 SEC.
compiling local iWeierstrassPGrad3 : List F -> F
Time: 0 SEC.
compiling local iWeierstrassPPrimeGrad1 : List F -> F
Time: 0.58 SEC.
compiling local iWeierstrassPPrimeGrad2 : List F -> F
Time: 0.20 SEC.
compiling local iWeierstrassPPrimeGrad3 : List F -> F
Time: 0.01 SEC.
compiling local iWeierstrassSigmaGrad1 : List F -> F
Time: 1.15 SEC.
compiling local iWeierstrassSigmaGrad2 : List F -> F
Time: 0.24 SEC.
compiling local iWeierstrassSigmaGrad3 : List F -> F
Time: 0 SEC.
compiling local iWeierstrassZetaGrad1 : List F -> F
Time: 0.19 SEC.
compiling local iWeierstrassZetaGrad2 : List F -> F
Time: 0.14 SEC.
compiling local iWeierstrassZetaGrad3 : List F -> F
Time: 0 SEC.
processing macro definition OF ==> OutputForm
processing macro definition SEX ==> SExpression
processing macro definition NNI ==> NonNegativeInteger
****** Domain: F already in scope
augmenting F: (RetractableTo (Integer))
augmenting $: (SIGNATURE $ hypergeometricF (F (List F) (List F) F))
augmenting $: (SIGNATURE $ meijerG (F (List F) (List F) (List F) (List F) F))
compiling local get_int_listf : List F -> List Integer
Time: 0.01 SEC.
compiling local replace_i : (List F,
compiling exported iiHypergeometricF : List F -> F
Time: 0.02 SEC.
compiling local idvsum : (BasicOperator,
compiling local dvhypergeom : (List F,
compiling local add_pairs_to_list : (List List F,
compiling local dvmeijer : (List F,
compiling local get_if_list : (Integer,
compiling local get_if_lists : (List Integer,
compiling local get_int_listi : (Integer,
compiling local get_of_list : (Integer,
compiling local get_of_lists : (List Integer,
compiling local get_int_listo : (Integer,
compiling local dhyper0 : (OutputForm,
compiling local dhyper : List OutputForm -> OutputForm
Time: 0 SEC.
compiling local ddhyper : List OutputForm -> OutputForm
Time: 0 SEC.
compiling local dmeijer0 : (OutputForm,
compiling local dmeijer : List OutputForm -> OutputForm
Time: 0 SEC.
compiling local ddmeijer : List OutputForm -> OutputForm
Time: 0 SEC.
compiling local inhyper : List InputForm -> InputForm
Time: 0.01 SEC.
compiling local inmeijer : List InputForm -> InputForm
Time: 0 SEC.
compiling exported iiHypergeometricF : List F -> F
Time: 0.01 SEC.
compiling local iiMeijerG : List F -> F
Time: 0 SEC.
compiling local d_eis : F -> F
Time: 0 SEC.
****** Domain: F already in scope
augmenting F: (TranscendentalFunctionCategory)
****** Domain: F already in scope
augmenting F: (RadicalCategory)
compiling local d_erfs : F -> F
Time: 0.01 SEC.
compiling local d_erfis : F -> F
Time: 0.01 SEC.
****** Domain: F already in scope
augmenting F: (RetractableTo (Integer))
augmenting $: (SIGNATURE $ hypergeometricF (F (List F) (List F) F))
augmenting $: (SIGNATURE $ meijerG (F (List F) (List F) (List F) (List F) F))
(time taken in buildFunctor: 10)
;;; *** |FunctionalSpecialFunction| REDEFINED
;;; *** |FunctionalSpecialFunction| REDEFINED
Time: 0.07 SEC.
Warnings:
[1] iiabs: not known that (Ring) is of mode (CATEGORY R (SIGNATURE abs (R R)))
[2] iiconjugate: not known that (Ring) is of mode (CATEGORY R (SIGNATURE conjugate (R R)))
Cumulative Statistics for Constructor FunctionalSpecialFunction
Time: 5.71 seconds
finalizing NRLIB FSPECX
Processing FunctionalSpecialFunction for Browser database:
--------constructor---------
--------(belong? ((Boolean) (BasicOperator)))---------
--------(operator ((BasicOperator) (BasicOperator)))---------
--------(abs (F F))---------
--------(conjugate (F F))---------
--------(Gamma (F F))---------
--------(Gamma (F F F))---------
--------(Beta (F F F))---------
--------(digamma (F F))---------
--------(polygamma (F F F))---------
--------(besselJ (F F F))---------
--------(besselY (F F F))---------
--------(besselI (F F F))---------
--------(besselK (F F F))---------
--------(airyAi (F F))---------
--------(airyAiPrime (F F))---------
--------(airyBi (F F))---------
--------(airyBiPrime (F F))---------
--------(lambertW (F F))---------
--------(polylog (F F F))---------
--------(weierstrassP (F F F F))---------
--------(weierstrassPPrime (F F F F))---------
--------(weierstrassSigma (F F F F))---------
--------(weierstrassZeta (F F F F))---------
--------(whittakerM (F F F F))---------
--------(whittakerW (F F F F))---------
--------(angerJ (F F F))---------
--------(weberE (F F F))---------
--------(struveH (F F F))---------
--------(struveL (F F F))---------
--------(hankelH1 (F F F))---------
--------(hankelH2 (F F F))---------
--------(lommelS1 (F F F F))---------
--------(lommelS2 (F F F F))---------
--------(kummerM (F F F F))---------
--------(kummerU (F F F F))---------
--------(legendreP (F F F F))---------
--------(legendreQ (F F F F))---------
--------(kelvinBei (F F F))---------
--------(kelvinBer (F F F))---------
--------(kelvinKei (F F F))---------
--------(kelvinKer (F F F))---------
--------(ellipticK (F F))---------
--------(ellipticE (F F))---------
--------(ellipticE (F F F))---------
--------(ellipticF (F F F))---------
--------(ellipticPi (F F F F))---------
--------(jacobiSn (F F F))---------
--------(jacobiCn (F F F))---------
--------(jacobiDn (F F F))---------
--------(jacobiZeta (F F F))---------
--------(jacobiTheta (F F F))---------
--------(lerchPhi (F F F F))---------
--------(riemannZeta (F F))---------
--------(charlierC (F F F F))---------
--------(hermiteH (F F F))---------
--------(jacobiP (F F F F F))---------
--------(laguerreL (F F F F))---------
--------(meixnerM (F F F F F))---------
--------(hypergeometricF (F (List F) (List F) F))---------
--------(meijerG (F (List F) (List F) (List F) (List F) F))---------
--------(iiGamma (F F))---------
--------(iiabs (F F))---------
--------(iiconjugate (F F))---------
--------(iiBeta (F (List F)))---------
--------(iidigamma (F F))---------
--------(iipolygamma (F (List F)))---------
--------(iiBesselJ (F (List F)))---------
--------(iiBesselY (F (List F)))---------
--------(iiBesselI (F (List F)))---------
--------(iiBesselK (F (List F)))---------
--------(iiAiryAi (F F))---------
--------(iiAiryAiPrime (F F))---------
--------(iiAiryBi (F F))---------
--------(iiAiryBiPrime (F F))---------
--------(iAiryAi (F F))---------
--------(iAiryAiPrime (F F))---------
--------(iAiryBi (F F))---------
--------(iAiryBiPrime (F F))---------
--------(iiHypergeometricF (F (List F)))---------
--------(iiPolylog (F F F))---------
--------(iLambertW (F F))---------
; compiling file "/var/aw/var/LatexWiki/FSPECX.NRLIB/FSPECX.lsp" (written 15 AUG 2014 05:30:54 PM):
; /var/aw/var/LatexWiki/FSPECX.NRLIB/FSPECX.fasl written
; compilation finished in 0:00:05.937
------------------------------------------------------------------------
FunctionalSpecialFunction is now explicitly exposed in frame initial
FunctionalSpecialFunction will be automatically loaded when needed
from /var/aw/var/LatexWiki/FSPECX.NRLIB/FSPECXfricas
differentiate(abs(x),x)
 | (1) |
Type: Expression(Integer)
fricas
differentiate(conjugate(x),x)
 | (2) |
Type: Expression(Integer)
fricas
conjugate(x)
 | (3) |
Type: Expression(Integer)
fricas
conjugate(z::Expression Complex INT)
 | (4) |
Type: Expression(Complex(Integer))
fricas
conjugate(conjugate(z::Expression Complex INT))
 | (5) |
Type: Expression(Complex(Integer))
fricas
conjugate((1+%i)::Expression Complex INT)
 | (6) |
Type: Expression(Complex(Integer))
fricas
conjugate(a+%i*b)
 | (7) |
Type: Expression(Complex(Integer))