15.1: Difference between revisions

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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/15.1.E1 15.1.E1] || [[Item:Q4975|<math>\genhyperF{2}{1}@{a,b}{c}{z} = \hyperF@{a}{b}{c}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{2}{1}@{a,b}{c}{z} = \hyperF@{a}{b}{c}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>hypergeom([a , b], [c], z) = hypergeom([a, b], [c], z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{a , b}, {c}, z] == Hypergeometric2F1[a, b, c, z]</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
| [https://dlmf.nist.gov/15.1.E1 15.1.E1] || <math qid="Q4975">\genhyperF{2}{1}@{a,b}{c}{z} = \hyperF@{a}{b}{c}{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{2}{1}@{a,b}{c}{z} = \hyperF@{a}{b}{c}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>hypergeom([a , b], [c], z) = hypergeom([a, b], [c], z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{a , b}, {c}, z] == Hypergeometric2F1[a, b, c, z]</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, 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/15.1.E1 15.1.E1] || [[Item:Q4975|<math>\hyperF@{a}{b}{c}{z} = \hyperF@@{a}{b}{c}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\hyperF@{a}{b}{c}{z} = \hyperF@@{a}{b}{c}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>hypergeom([a, b], [c], z) = hypergeom([a, b], [c], z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric2F1[a, b, c, z] == Hypergeometric2F1[a, b, c, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 300]
| [https://dlmf.nist.gov/15.1.E1 15.1.E1] || <math qid="Q4975">\hyperF@{a}{b}{c}{z} = \hyperF@@{a}{b}{c}{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\hyperF@{a}{b}{c}{z} = \hyperF@@{a}{b}{c}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>hypergeom([a, b], [c], z) = hypergeom([a, b], [c], z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric2F1[a, b, c, z] == Hypergeometric2F1[a, b, c, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 300]
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| [https://dlmf.nist.gov/15.1.E2 15.1.E2] || [[Item:Q4976|<math>\frac{\hyperF@{a}{b}{c}{z}}{\EulerGamma@{c}} = \hyperOlverF@{a}{b}{c}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\hyperF@{a}{b}{c}{z}}{\EulerGamma@{c}} = \hyperOlverF@{a}{b}{c}{z}</syntaxhighlight> || <math>\realpart@@{c} > 0, |z| < 1</math> || <syntaxhighlight lang=mathematica>(hypergeom([a, b], [c], z))/(GAMMA(c)) = hypergeom([a, b], [c], z)/GAMMA(c)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Hypergeometric2F1[a, b, c, z],Gamma[c]] == Hypergeometric2F1Regularized[a, b, c, z]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 108]
| [https://dlmf.nist.gov/15.1.E2 15.1.E2] || <math qid="Q4976">\frac{\hyperF@{a}{b}{c}{z}}{\EulerGamma@{c}} = \hyperOlverF@{a}{b}{c}{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\hyperF@{a}{b}{c}{z}}{\EulerGamma@{c}} = \hyperOlverF@{a}{b}{c}{z}</syntaxhighlight> || <math>\realpart@@{c} > 0, |z| < 1</math> || <syntaxhighlight lang=mathematica>(hypergeom([a, b], [c], z))/(GAMMA(c)) = hypergeom([a, b], [c], z)/GAMMA(c)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Hypergeometric2F1[a, b, c, z],Gamma[c]] == Hypergeometric2F1Regularized[a, b, c, z]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 108]
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| [https://dlmf.nist.gov/15.1.E2 15.1.E2] || [[Item:Q4976|<math>\hyperOlverF@{a}{b}{c}{z} = \hyperOlverF@@{a}{b}{c}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\hyperOlverF@{a}{b}{c}{z} = \hyperOlverF@@{a}{b}{c}{z}</syntaxhighlight> || <math>\realpart@@{c} > 0, |z| < 1</math> || <syntaxhighlight lang=mathematica>hypergeom([a, b], [c], z)/GAMMA(c) = hypergeom([a, b], [c], z)/GAMMA(c)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric2F1Regularized[a, b, c, z] == Hypergeometric2F1Regularized[a, b, c, z]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 108]
| [https://dlmf.nist.gov/15.1.E2 15.1.E2] || <math qid="Q4976">\hyperOlverF@{a}{b}{c}{z} = \hyperOlverF@@{a}{b}{c}{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\hyperOlverF@{a}{b}{c}{z} = \hyperOlverF@@{a}{b}{c}{z}</syntaxhighlight> || <math>\realpart@@{c} > 0, |z| < 1</math> || <syntaxhighlight lang=mathematica>hypergeom([a, b], [c], z)/GAMMA(c) = hypergeom([a, b], [c], z)/GAMMA(c)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric2F1Regularized[a, b, c, z] == Hypergeometric2F1Regularized[a, b, c, z]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 108]
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| [https://dlmf.nist.gov/15.1.E2 15.1.E2] || [[Item:Q4976|<math>\hyperOlverF@@{a}{b}{c}{z} = \genhyperOlverF{2}{1}@{a,b}{c}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\hyperOlverF@@{a}{b}{c}{z} = \genhyperOlverF{2}{1}@{a,b}{c}{z}</syntaxhighlight> || <math>\realpart@@{c} > 0, |z| < 1</math> || <syntaxhighlight lang=mathematica>hypergeom([a, b], [c], z)/GAMMA(c) = hypergeom([a , b], [c], z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric2F1Regularized[a, b, c, z] == HypergeometricPFQRegularized[{a , b}, {c}, z]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [175 / 216]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.2039500354
| [https://dlmf.nist.gov/15.1.E2 15.1.E2] || <math qid="Q4976">\hyperOlverF@@{a}{b}{c}{z} = \genhyperOlverF{2}{1}@{a,b}{c}{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\hyperOlverF@@{a}{b}{c}{z} = \genhyperOlverF{2}{1}@{a,b}{c}{z}</syntaxhighlight> || <math>\realpart@@{c} > 0, |z| < 1</math> || <syntaxhighlight lang=mathematica>hypergeom([a, b], [c], z)/GAMMA(c) = hypergeom([a , b], [c], z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Hypergeometric2F1Regularized[a, b, c, z] == HypergeometricPFQRegularized[{a , b}, {c}, z]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [175 / 216]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.2039500354
Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .227101342
Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .227101342
Test Values: {a = -3/2, b = -3/2, c = 3/2, z = 1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 108]
Test Values: {a = -3/2, b = -3/2, c = 3/2, z = 1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 108]
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Latest revision as of 11:38, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
15.1.E1 F 1 2 ⁡ ( a , b ; c ; z ) = F ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(a,b;c;z\right)=F\left(a,b;c;z% \right)}}
\genhyperF{2}{1}@{a,b}{c}{z} = \hyperF@{a}{b}{c}{z}		 

hypergeom([a , b], [c], z) = hypergeom([a, b], [c], z)

HypergeometricPFQ[{a , b}, {c}, z] == Hypergeometric2F1[a, b, c, z]
Successful Successful -
Failed [42 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
15.1.E1 F ⁡ ( a , b ; c ; z ) = F ⁡ ( a , b c ; z ) Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle F\left(a,b;c;z\right)=F\left({a,b\atop c};z\right% )}}
\hyperF@{a}{b}{c}{z} = \hyperF@@{a}{b}{c}{z}		 

hypergeom([a, b], [c], z) = hypergeom([a, b], [c], z)

Hypergeometric2F1[a, b, c, z] == Hypergeometric2F1[a, b, c, z]
Successful Successful - Successful [Tested: 300]
15.1.E2 F ⁡ ( a , b ; c ; z ) Γ ⁡ ( c ) = 𝐅 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 Euler-Gamma 𝑐 scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}% =\mathbf{F}\left(a,b;c;z\right)}}
\frac{\hyperF@{a}{b}{c}{z}}{\EulerGamma@{c}} = \hyperOlverF@{a}{b}{c}{z}		 ℜ ⁡ c > 0 , | z | < 1 formulae-sequence 𝑐 0 𝑧 1 {\displaystyle{\displaystyle\Re c>0,|z|<1}} 
(hypergeom([a, b], [c], z))/(GAMMA(c)) = hypergeom([a, b], [c], z)/GAMMA(c)

Divide[Hypergeometric2F1[a, b, c, z],Gamma[c]] == Hypergeometric2F1Regularized[a, b, c, z]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 108]
15.1.E2 𝐅 ⁡ ( a , b ; c ; z ) = 𝐅 ⁡ ( a , b c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\mathbf{F}\left({a,% b\atop c};z\right)}}
\hyperOlverF@{a}{b}{c}{z} = \hyperOlverF@@{a}{b}{c}{z}		 ℜ ⁡ c > 0 , | z | < 1 formulae-sequence 𝑐 0 𝑧 1 {\displaystyle{\displaystyle\Re c>0,|z|<1}} 
hypergeom([a, b], [c], z)/GAMMA(c) = hypergeom([a, b], [c], z)/GAMMA(c)

Hypergeometric2F1Regularized[a, b, c, z] == Hypergeometric2F1Regularized[a, b, c, z]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 108]
15.1.E2 𝐅 ⁡ ( a , b c ; z ) = 𝐅 1 2 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 hypergeometric-bold-pFq 2 1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop c};z\right)={{}_{2}{% \mathbf{F}}_{1}}\left(a,b;c;z\right)}}
\hyperOlverF@@{a}{b}{c}{z} = \genhyperOlverF{2}{1}@{a,b}{c}{z}		 ℜ ⁡ c > 0 , | z | < 1 formulae-sequence 𝑐 0 𝑧 1 {\displaystyle{\displaystyle\Re c>0,|z|<1}} 
hypergeom([a, b], [c], z)/GAMMA(c) = hypergeom([a , b], [c], z)

Hypergeometric2F1Regularized[a, b, c, z] == HypergeometricPFQRegularized[{a , b}, {c}, z]
Failure Successful
Failed [175 / 216]
Result: -.2039500354
Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/2}


Result: .227101342
Test Values: {a = -3/2, b = -3/2, c = 3/2, z = 1/2}

... skip entries to safe data
 Successful [Tested: 108]
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