15.14: 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/15.14.E1 15.14.E1] || [[Item:Q5176|<math>\int_{0}^{\infty}x^{s-1}\hyperOlverF@@{a}{b}{c}{-x}\diff{x} = \frac{\EulerGamma@{s}\EulerGamma@{a-s}\EulerGamma@{b-s}}{\EulerGamma@{a}\EulerGamma@{b}\EulerGamma@{c-s}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}x^{s-1}\hyperOlverF@@{a}{b}{c}{-x}\diff{x} = \frac{\EulerGamma@{s}\EulerGamma@{a-s}\EulerGamma@{b-s}}{\EulerGamma@{a}\EulerGamma@{b}\EulerGamma@{c-s}}</syntaxhighlight> || <math>\min(\realpart@@{a} > \realpart@@{s}, \realpart@@{b}) > \realpart@@{s}, \realpart@@{s} > 0, \realpart@@{(a-s)} > 0, \realpart@@{(b-s)} > 0, \realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{(c-s)} > 0, |(-x)| < 1, \realpart@@{(c+s)} > 0</math> || <syntaxhighlight lang=mathematica>int((x)^(s - 1)* hypergeom([a, b], [c], - x)/GAMMA(c), x = 0..infinity) = (GAMMA(s)*GAMMA(a - s)*GAMMA(b - s))/(GAMMA(a)*GAMMA(b)*GAMMA(c - s))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(x)^(s - 1)* Hypergeometric2F1Regularized[a, b, c, - x], {x, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[s]*Gamma[a - s]*Gamma[b - s],Gamma[a]*Gamma[b]*Gamma[c - s]]</syntaxhighlight> || Successful || Aborted || - || Skipped - Because timed out
| [https://dlmf.nist.gov/15.14.E1 15.14.E1] || <math qid="Q5176">\int_{0}^{\infty}x^{s-1}\hyperOlverF@@{a}{b}{c}{-x}\diff{x} = \frac{\EulerGamma@{s}\EulerGamma@{a-s}\EulerGamma@{b-s}}{\EulerGamma@{a}\EulerGamma@{b}\EulerGamma@{c-s}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}x^{s-1}\hyperOlverF@@{a}{b}{c}{-x}\diff{x} = \frac{\EulerGamma@{s}\EulerGamma@{a-s}\EulerGamma@{b-s}}{\EulerGamma@{a}\EulerGamma@{b}\EulerGamma@{c-s}}</syntaxhighlight> || <math>\min(\realpart@@{a} > \realpart@@{s}, \realpart@@{b}) > \realpart@@{s}, \realpart@@{s} > 0, \realpart@@{(a-s)} > 0, \realpart@@{(b-s)} > 0, \realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{(c-s)} > 0, |(-x)| < 1, \realpart@@{(c+s)} > 0</math> || <syntaxhighlight lang=mathematica>int((x)^(s - 1)* hypergeom([a, b], [c], - x)/GAMMA(c), x = 0..infinity) = (GAMMA(s)*GAMMA(a - s)*GAMMA(b - s))/(GAMMA(a)*GAMMA(b)*GAMMA(c - s))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(x)^(s - 1)* Hypergeometric2F1Regularized[a, b, c, - x], {x, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[s]*Gamma[a - s]*Gamma[b - s],Gamma[a]*Gamma[b]*Gamma[c - s]]</syntaxhighlight> || Successful || Aborted || - || Skipped - Because timed out
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Latest revision as of 11:41, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
15.14.E1 ∫ 0 ∞ x s - 1 ⁢ 𝐅 ⁡ ( a , b c ; - x ) ⁢ d x = Γ ⁡ ( s ) ⁢ Γ ⁡ ( a - s ) ⁢ Γ ⁡ ( b - s ) Γ ⁡ ( a ) ⁢ Γ ⁡ ( b ) ⁢ Γ ⁡ ( c - s ) superscript subscript 0 superscript 𝑥 𝑠 1 scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑥 𝑥 Euler-Gamma 𝑠 Euler-Gamma 𝑎 𝑠 Euler-Gamma 𝑏 𝑠 Euler-Gamma 𝑎 Euler-Gamma 𝑏 Euler-Gamma 𝑐 𝑠 {\displaystyle{\displaystyle\int_{0}^{\infty}x^{s-1}\mathbf{F}\left({a,b\atop c% };-x\right)\mathrm{d}x=\frac{\Gamma\left(s\right)\Gamma\left(a-s\right)\Gamma% \left(b-s\right)}{\Gamma\left(a\right)\Gamma\left(b\right)\Gamma\left(c-s% \right)}}}
\int_{0}^{\infty}x^{s-1}\hyperOlverF@@{a}{b}{c}{-x}\diff{x} = \frac{\EulerGamma@{s}\EulerGamma@{a-s}\EulerGamma@{b-s}}{\EulerGamma@{a}\EulerGamma@{b}\EulerGamma@{c-s}}		 min ⁡ ( ℜ ⁡ a > ℜ ⁡ s , ℜ ⁡ b ) > ℜ ⁡ s , ℜ ⁡ s > 0 , ℜ ⁡ ( a - s ) > 0 , ℜ ⁡ ( b - s ) > 0 , ℜ ⁡ a > 0 , ℜ ⁡ b > 0 , ℜ ⁡ ( c - s ) > 0 , | ( - x ) | < 1 , ℜ ⁡ ( c + s ) > 0 formulae-sequence 𝑎 𝑠 𝑏 𝑠 formulae-sequence 𝑠 0 formulae-sequence 𝑎 𝑠 0 formulae-sequence 𝑏 𝑠 0 formulae-sequence 𝑎 0 formulae-sequence 𝑏 0 formulae-sequence 𝑐 𝑠 0 formulae-sequence 𝑥 1 𝑐 𝑠 0 {\displaystyle{\displaystyle\min(\Re a>\Re s,\Re b)>\Re s,\Re s>0,\Re(a-s)>0,% \Re(b-s)>0,\Re a>0,\Re b>0,\Re(c-s)>0,|(-x)|<1,\Re(c+s)>0}} 
int((x)^(s - 1)* hypergeom([a, b], [c], - x)/GAMMA(c), x = 0..infinity) = (GAMMA(s)*GAMMA(a - s)*GAMMA(b - s))/(GAMMA(a)*GAMMA(b)*GAMMA(c - s))

Integrate[(x)^(s - 1)* Hypergeometric2F1Regularized[a, b, c, - x], {x, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[s]*Gamma[a - s]*Gamma[b - s],Gamma[a]*Gamma[b]*Gamma[c - s]]
Successful Aborted - Skipped - Because timed out
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