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SandBoxDoublePowerSeries

Edit detail for SandBoxDoublePowerSeries revision 1 of 19

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Editor: Bill Page Time: 2014/08/20 23:34:39 GMT+0
Note:

changed:
-
\begin{axiom}
a!
!:=operator '!
a(i,j)==(1+(j-1)*i)/!(2*i+4*j)
a0:=matrix [[a(i,j)*k^i*l^j for i in 0..5] for j in 1..4]
a(i,j)==(1+(j-1)*i)/Gamma(1+2*i+4*j)
a0:=matrix [[a(i,j)*k^i*l^j for i in 0..5] for j in 1..4]
i0:=a(1/2+%i*beta,0)*(-x)^(1/2+%i*beta)/cosh(%pi*beta)
i1:=a(0,1/2+%i*delta)*(-x)^(1/2+%i*beta)*(-y)^(1/2+%i*delta)/cosh(%pi*delta)
i2:=a(1/2+%i*beta,1/2+%i*delta)*(-y)^(1/2+%i*delta)/(cosh(%pi*beta)*cosh(%pi*delta))
eval(i0,beta=-1)
eval(i0,beta=0)
eval(i0,beta=1)
eval(i1,delta=-1)
eval(i1,delta=0)
eval(i1,delta=1)
\end{axiom}

\begin{axiom}
integrate(i0,beta=%minusInfinity..%plusInfinity,"noPole")
\end{axiom}

\begin{axiom}
integrate(i1,delta=%minusInfinity..%plusInfinity,"noPole")
\end{axiom}

fricas

a!

$\label{eq1}a !$

(1)

Type: Variable(a!)

fricas

!:=operator '!

$\label{eq2}!$

(2)

Type: BasicOperator?

fricas

a(i,j)==(1+(j-1)*i)/!(2*i+4*j)

Type: Void

fricas

a0:=matrix [[a(i,j)*k^i*l^j for i in 0..5] for j in 1..4]

fricas

Compiling function a with type (NonNegativeInteger,PositiveInteger)
       -> Expression(Integer)

$\label{eq3}\left[ \begin{array}{cccccc} {l \over{! \left({4}\right)}}&{{k \ l}\over{! \left({6}\right)}}&{{{{k}^{2}}\ l}\over{! \left({8}\right)}}&{{{{k}^{3}}\ l}\over{! \left({1 0}\right)}}&{{{{k}^{4}}\ l}\over{! \left({12}\right)}}&{{{{k}^{5}}\ l}\over{! \left({14}\right)}} \ {{{l}^{2}}\over{! \left({8}\right)}}&{{2 \ k \ {{l}^{2}}}\over{! \left({10}\right)}}&{{3 \ {{k}^{2}}\ {{l}^{2}}}\over{! \left({1 2}\right)}}&{{4 \ {{k}^{3}}\ {{l}^{2}}}\over{! \left({14}\right)}}&{{5 \ {{k}^{4}}\ {{l}^{2}}}\over{! \left({16}\right)}}&{{6 \ {{k}^{5}}\ {{l}^{2}}}\over{! \left({18}\right)}} \ {{{l}^{3}}\over{! \left({12}\right)}}&{{3 \ k \ {{l}^{3}}}\over{! \left({14}\right)}}&{{5 \ {{k}^{2}}\ {{l}^{3}}}\over{! \left({1 6}\right)}}&{{7 \ {{k}^{3}}\ {{l}^{3}}}\over{! \left({18}\right)}}&{{9 \ {{k}^{4}}\ {{l}^{3}}}\over{! \left({20}\right)}}&{{{11}\ {{k}^{5}}\ {{l}^{3}}}\over{! \left({22}\right)}} \ {{{l}^{4}}\over{! \left({16}\right)}}&{{4 \ k \ {{l}^{4}}}\over{! \left({18}\right)}}&{{7 \ {{k}^{2}}\ {{l}^{4}}}\over{! \left({2 0}\right)}}&{{{10}\ {{k}^{3}}\ {{l}^{4}}}\over{! \left({22}\right)}}&{{{1 3}\ {{k}^{4}}\ {{l}^{4}}}\over{! \left({24}\right)}}&{{{16}\ {{k}^{5}}\ {{l}^{4}}}\over{! \left({26}\right)}}$

(3)

Type: Matrix(Expression(Integer))

fricas

a(i,j)==(1+(j-1)*i)/Gamma(1+2*i+4*j)

   Compiled code for a has been cleared.
   1 old definition(s) deleted for function or rule a

Type: Void

fricas

a0:=matrix [[a(i,j)*k^i*l^j for i in 0..5] for j in 1..4]

fricas

Compiling function a with type (NonNegativeInteger,PositiveInteger)
       -> Expression(Integer)

$\label{eq4}\left[ \begin{array}{cccccc} {l \over{24}}&{{k \ l}\over{720}}&{{{{k}^{2}}\ l}\over{4032 0}}&{{{{k}^{3}}\ l}\over{3628800}}&{{{{k}^{4}}\ l}\over{479 001600}}&{{{{k}^{5}}\ l}\over{87178291200}} \ {{{l}^{2}}\over{40320}}&{{k \ {{l}^{2}}}\over{1814400}}&{{{{k}^{2}}\ {{l}^{2}}}\over{159667200}}&{{{{k}^{3}}\ {{l}^{2}}}\over{2 1794572800}}&{{{{k}^{4}}\ {{l}^{2}}}\over{4184557977600}}&{{{{k}^{5}}\ {{l}^{2}}}\over{1067062284288000}} \ {{{l}^{3}}\over{479001600}}&{{k \ {{l}^{3}}}\over{29059430400}}&{{{{k}^{2}}\ {{l}^{3}}}\over{4184557977600}}&{{{{k}^{3}}\ {{l}^{3}}}\over{9 14624815104000}}&{{{{k}^{4}}\ {{l}^{3}}}\over{270322445352960 000}}&{{{{k}^{5}}\ {{l}^{3}}}\over{102181884343418880000}} \ {{{l}^{4}}\over{20922789888000}}&{{k \ {{l}^{4}}}\over{160059 3426432000}}&{{{{k}^{2}}\ {{l}^{4}}}\over{347557429739520000}}&{{{{k}^{3}}\ {{l}^{4}}}\over{112400072777760768000}}&{{{{k}^{4}}\ {{l}^{4}}}\over{47726800133326110720000}}&{{{{k}^{5}}\ {{l}^{4}}}\over{2 5205716320412852224000000}}$

(4)

Type: Matrix(Expression(Integer))

fricas

i0:=a(1/2+%i*beta,0)*(-x)^(1/2+%i*beta)/cosh(%pi*beta)

fricas

Compiling function a with type (Polynomial(Fraction(Complex(Integer)
      )),NonNegativeInteger) -> Expression(Complex(Integer))

$\label{eq5}{{\left(-{2 \ i \ beta}+ 1 \right)}\ {{\left(- x \right)}^{{{2 \ i \ beta}+ 1}\over 2}}}\over{2 \ {\Gamma \left({{2 \ i \ beta}+ 2}\right)}\ {\cosh \left({beta \ \pi}\right)}}$

(5)

Type: Expression(Complex(Integer))

fricas

i1:=a(0,1/2+%i*delta)*(-x)^(1/2+%i*beta)*(-y)^(1/2+%i*delta)/cosh(%pi*delta)

fricas

Compiling function a with type (NonNegativeInteger,Polynomial(
      Fraction(Complex(Integer)))) -> Expression(Complex(Integer))

$\label{eq6}{{{\left(- y \right)}^{{{2 \ i \ delta}+ 1}\over 2}}\ {{\left(- x \right)}^{{{2 \ i \ beta}+ 1}\over 2}}}\over{{\Gamma \left({{4 \ i \ delta}+ 3}\right)}\ {\cosh \left({delta \ \pi}\right)}}$

(6)

Type: Expression(Complex(Integer))

fricas

i2:=a(1/2+%i*beta,1/2+%i*delta)*(-y)^(1/2+%i*delta)/(cosh(%pi*beta)*cosh(%pi*delta))

fricas

Compiling function a with type (Polynomial(Fraction(Complex(Integer)
      )),Polynomial(Fraction(Complex(Integer)))) -> Expression(Complex(
      Integer))

$\label{eq7}{\left({\left({ \begin{array}{@{}l} \displaystyle {{\left(-{4 \ beta}+{2 \ i}\right)}\ delta}- \ \ \displaystyle {2 \ i \ beta}+ 3$

(7)

Type: Expression(Complex(Integer))

fricas

eval(i0,beta=-1)

$\label{eq8}{{\left(1 +{2 \ i}\right)}\ {{\left(- x \right)}^{{1 -{2 \ i}}\over 2}}}\over{2 \ {\Gamma \left({2 -{2 \ i}}\right)}\ {\cosh \left({\pi}\right)}}$

(8)

Type: Expression(Complex(Integer))

fricas

eval(i0,beta=0)

$\label{eq9}{\sqrt{- x}}\over 2$

(9)

Type: Expression(Complex(Integer))

fricas

eval(i0,beta=1)

$\label{eq10}{{\left(1 -{2 \ i}\right)}\ {{\left(- x \right)}^{{1 +{2 \ i}}\over 2}}}\over{2 \ {\Gamma \left({2 +{2 \ i}}\right)}\ {\cosh \left({\pi}\right)}}$

(10)

Type: Expression(Complex(Integer))

fricas

eval(i1,delta=-1)

$\label{eq11}{{{\left(- y \right)}^{{1 -{2 \ i}}\over 2}}\ {{\left(- x \right)}^{{{2 \ i \ beta}+ 1}\over 2}}}\over{{\Gamma \left({3 -{4 \ i}}\right)}\ {\cosh \left({\pi}\right)}}$

(11)

Type: Expression(Complex(Integer))

fricas

eval(i1,delta=0)

$\label{eq12}{{{\left(- x \right)}^{{{2 \ i \ beta}+ 1}\over 2}}\ {\sqrt{- y}}}\over 2$

(12)

Type: Expression(Complex(Integer))

fricas

eval(i1,delta=1)

$\label{eq13}{{{\left(- y \right)}^{{1 +{2 \ i}}\over 2}}\ {{\left(- x \right)}^{{{2 \ i \ beta}+ 1}\over 2}}}\over{{\Gamma \left({3 +{4 \ i}}\right)}\ {\cosh \left({\pi}\right)}}$

(13)

Type: Expression(Complex(Integer))

fricas

integrate(i0,beta=%minusInfinity..%plusInfinity,"noPole")

   >> System error:
   arithmetic error DIVISION-BY-ZERO signalled
Operation was SB-KERNEL::DIVISION, operands (0 0).

fricas

integrate(i1,delta=%minusInfinity..%plusInfinity,"noPole")

   >> System error:
   arithmetic error DIVISION-BY-ZERO signalled
Operation was SB-KERNEL::DIVISION, operands (0 0).