View source for 23.17

{{DISPLAYTITLE:Weierstrass Elliptic and Modular Functions - 23.17 Elementary Properties}}

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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/23.17#Ex1 23.17#Ex1] || <math qid="Q7340">\modularlambdatau@{i} = \tfrac{1}{2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modularlambdatau@{i} = \tfrac{1}{2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>ModularLambda[I] == Divide[1,2]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/23.17#Ex3 23.17#Ex3] || <math qid="Q7342">\KleincompinvarJtau@{i} = 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\KleincompinvarJtau@{i} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>KleinInvariantJ[I] == 1</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/23.17#Ex5 23.17#Ex5] || <math qid="Q7344">\Dedekindeta@{i} = \frac{\EulerGamma@{\tfrac{1}{4}}}{2\pi^{3/4}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Dedekindeta@{i} = \frac{\EulerGamma@{\tfrac{1}{4}}}{2\pi^{3/4}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>DedekindEta[I] == Divide[Gamma[Divide[1,4]],2*(Pi)^(3/4)]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/23.17.E6 23.17.E6] || <math qid="Q7348">\Dedekindeta@{\tau} = \sum_{n=-\infty}^{\infty}(-1)^{n}q^{(6n+1)^{2}/12}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Dedekindeta@{\tau} = \sum_{n=-\infty}^{\infty}(-1)^{n}q^{(6n+1)^{2}/12}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>DedekindEta[\[Tau]] == Sum[(- 1)^(n)* (q)^((6*n + 1)^(2)/12), {n, - Infinity, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.7682254223260567, Times[-1.0, NSum[Times[Power[-1, n], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[Rational[1, 12], Power[Plus[1, Times[6, n]], 2]]]]
Test Values: {n, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Complex[0, 1]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[0.7682254223260567, Times[-1.0, NSum[Times[Power[-1, n], Power[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], Times[Rational[1, 12], Power[Plus[1, Times[6, n]], 2]]]]
Test Values: {n, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Complex[0, 1]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/23.17.E7 23.17.E7] || <math qid="Q7349">\modularlambdatau@{\tau} = 16q\prod_{n=1}^{\infty}\left(\frac{1+q^{2n}}{1+q^{2n-1}}\right)^{8}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modularlambdatau@{\tau} = 16q\prod_{n=1}^{\infty}\left(\frac{1+q^{2n}}{1+q^{2n-1}}\right)^{8}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>ModularLambda[\[Tau]] == 16*q*Product[(Divide[1 + (q)^(2*n),1 + (q)^(2*n - 1)])^(8), {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [24 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/23.17.E8 23.17.E8] || <math qid="Q7350">\Dedekindeta@{\tau} = q^{1/12}\prod_{n=1}^{\infty}(1-q^{2n})</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Dedekindeta@{\tau} = q^{1/12}\prod_{n=1}^{\infty}(1-q^{2n})</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>DedekindEta[\[Tau]] == (q)^(1/12)* Product[1 - (q)^(2*n), {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [4 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: DirectedInfinity[]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Complex[0, 1]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: DirectedInfinity[]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Rule[τ, Complex[0, 1]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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