View source for 4.22

{{DISPLAYTITLE:Elementary Functions - 4.22 Infinite Products and Partial Fractions}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/4.22.E1 4.22.E1] || <math qid="Q1748">\sin@@{z} = z\prod_{n=1}^{\infty}\left(1-\frac{z^{2}}{n^{2}\pi^{2}}\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sin@@{z} = z\prod_{n=1}^{\infty}\left(1-\frac{z^{2}}{n^{2}\pi^{2}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sin(z) = z*product(1 -((z)^(2))/((n)^(2)* (Pi)^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sin[z] == z*Product[1 -Divide[(z)^(2),(n)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/4.22.E2 4.22.E2] || <math qid="Q1749">\cos@@{z} = \prod_{n=1}^{\infty}\left(1-\frac{4z^{2}}{(2n-1)^{2}\pi^{2}}\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\cos@@{z} = \prod_{n=1}^{\infty}\left(1-\frac{4z^{2}}{(2n-1)^{2}\pi^{2}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>cos(z) = product(1 -(4*(z)^(2))/((2*n - 1)^(2)* (Pi)^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Cos[z] == Product[1 -Divide[4*(z)^(2),(2*n - 1)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/4.22.E3 4.22.E3] || <math qid="Q1750">\cot@@{z} = \frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{z^{2}-n^{2}\pi^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\cot@@{z} = \frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{z^{2}-n^{2}\pi^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>cot(z) = (1)/(z)+ 2*z*sum((1)/((z)^(2)- (n)^(2)* (Pi)^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Cot[z] == Divide[1,z]+ 2*z*Sum[Divide[1,(z)^(2)- (n)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
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| [https://dlmf.nist.gov/4.22.E4 4.22.E4] || <math qid="Q1751">\csc^{2}@@{z} = \sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi)^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\csc^{2}@@{z} = \sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi)^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(csc(z))^(2) = sum((1)/((z - n*Pi)^(2)), n = - infinity..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Csc[z])^(2) == Sum[Divide[1,(z - n*Pi)^(2)], {n, - Infinity, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
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| [https://dlmf.nist.gov/4.22.E5 4.22.E5] || <math qid="Q1752">\csc@@{z} = \frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}-n^{2}\pi^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\csc@@{z} = \frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}-n^{2}\pi^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>csc(z) = (1)/(z)+ 2*z*sum(((- 1)^(n))/((z)^(2)- (n)^(2)* (Pi)^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Csc[z] == Divide[1,z]+ 2*z*Sum[Divide[(- 1)^(n),(z)^(2)- (n)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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