Hypergeometric2F1

Generated from the engine's own definition: compute-engine's symbol, which we neither extend nor document by hand.

Gauss hypergeometric function ₂F₁(a, b; c; z).

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaHypergeometric function↗MathWorldHypergeometricFunction↗WikidataQ21028472↗Encyclopedia of MathematicsHypergeometric_function↗Britannicascience/hypergeometric-function↗DLMF15.2↗FungrimHypergeometric2F1!↗Fungrim entryWolfram LanguageHypergeometric2F1!↗

Hypergeometric2F1(complex | infinity, complex | infinity, complex | infinity, complex | infinity) -> numberGauss hypergeometric function ₂F₁(a, b; c; z).

SageMathhypergeometric([complex | infinity, complex | infinity], [complex | infinity], complex | infinity)mpmathhyp2f1(complex | infinity, complex | infinity, complex | infinity, complex | infinity)Juliaenumeratio_ball(hypergeometric_2f1, complex | infinity, complex | infinity, complex | infinity, complex | infinity)

Typed signatures

Type	Package	Applies
(complex | infinity, complex | infinity, complex | infinity, complex | infinity) -> number	analyticreplaces compute-engine	always
(complex | infinity, complex | infinity, complex | infinity, complex | infinity) -> number	compute-engine	always

Domain: Compute engine

Details

The Gauss hypergeometric function ${}_2F_1(a,b;c;z) = \sum_{s\ge0} (a)_s(b)_s/(c)_s \cdot z^s/s!$ (DLMF 15.2.1) converges on the disc $|z| < 1$ and continues analytically elsewhere, with Euler's integral $\dfrac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_0^1 t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}\,dt$ for $\operatorname{Re}(c) > \operatorname{Re}(b) > 0$ (DLMF 15.6.1); see Gamma, Beta. It is the $p=2,q=1$ case of HypergeometricPFQ, and dividing it by $\Gamma(c)$ gives Hypergeometric2F1Regularized, entire in $c$ where this itself has poles. It is compute-engine's own native head, declared for real and complex $z$, including some continuation past $|z|=1$.

• ${}_2F_1(a,b;c;0) = 1$ for every $a, b, c$.
• $a$ or $b$ a nonpositive integer terminates the series into a polynomial in $z$.
• $c = b$ collapses to an elementary power: ${}_2F_1(a,b;b;z) = (1-z)^{-a}$ (DLMF 15.4.6).
• ${}_2F_1(1,1;2;z) = -z^{-1}\ln(1-z)$ (DLMF 15.4.1).
• The complete elliptic integral of the first kind: $K(k) = \dfrac{\pi}{2}\,{}_2F_1(\tfrac12,\tfrac12;1;k^2)$ (DLMF 15.9.24).
• Gauss's summation at $z=1$: ${}_2F_1(a,b;c;1) = \Gamma(c)\Gamma(c-a-b)/(\Gamma(c-a)\Gamma(c-b))$ for $\operatorname{Re}(c-a-b) > 0$ (DLMF 15.4.20).

Plot

the complex plot cannot lower Hypergeometric2F1

Examples

Basic #

The series is $1$ at the origin

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In$\mathrm{Hypergeometric2F1}(1, 1, 2, 0)$Out$1$

${}_2F_1(1, 1; 2; \tfrac12) = 2\ln 2$

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In$N(\mathrm{Hypergeometric2F1}(1, 1, 2, \frac{1}{2}), 20)$Out$1.386\,294\,361\,119\,890\,618\,8$

Known value · mpmath 1.3.0 hyp2f1

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In$\operatorname{Series}(\mathrm{Hypergeometric2F1}(a, b, c, x), x, 0, 3)$Out$1+x\mathrm{Hypergeometric2F1}^{(0, 0, 0, 1)}(a, b, c, 0)+\frac{1}{2}\mathrm{Hypergeometric2F1}^{(0, 0, 0, 2)}(a, b, c, 0)x^2+\frac{1}{6}\mathrm{Hypergeometric2F1}^{(0, 0, 0, 3)}(a, b, c, 0)x^3+O\left(x^4\right)$

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In$\operatorname{Normal}(\operatorname{Series}(\mathrm{Hypergeometric2F1}(2, 3, 4, x), x, ∞, 5))$Out$-\frac{9}{x^4}-\frac{6}{x^5}+\frac{3}{x^2}+(3(2\ln(-x)-1))/x^3$

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In$\mathrm{FullSimplify}(\operatorname{Series}(\mathrm{Hypergeometric2F1}(2, 3, 4, x), x, 1, 3))$Out$\operatorname{Series}(\mathrm{Hypergeometric2F1}(2, 3, 4, x), x, 1, 3)$

Scope #

live ↗

In$N(\mathrm{Hypergeometric2F1}(\frac{1}{2}, \frac{1}{3}, 2, 1), 50)$Out$1.159\,595\,266\,963\,928\,365\,769\,992\,051\,570\,020\,881\,945\,165\,263\,439\,8$

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In$\mathrm{Hypergeometric2F1}(\frac{1}{2}, 1, 1, \bigl\lbrack\bigl\lbrack\frac{1}{2}, 0\bigr\rbrack, \bigl\lbrack0, \frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\sqrt{2}, 1\bigr\rbrack, \bigl\lbrack1, \sqrt{2}\bigr\rbrack\bigr\rbrack$

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In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\mathrm{Hypergeometric2F1}(\frac{1}{2}, 1, 1, \operatorname{\_1}), \bigl\lbrack\bigl\lbrack\frac{1}{2}, 0\bigr\rbrack, \bigl\lbrack0, \frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\sqrt{2}, 0\bigr\rbrack, \bigl\lbrack0, \sqrt{2}\bigr\rbrack\bigr\rbrack$

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In$\operatorname{Normal}(\operatorname{Series}(\mathrm{Hypergeometric2F1}(1, 1, 2, x), x, ∞, 4))$Out$\frac{1}{3x^4}+\frac{1}{2x^3}+\frac{1}{x^2}-\frac{\ln(-x)}{x}$

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In$\operatorname{Normal}(\operatorname{Series}(\mathrm{Hypergeometric2F1}(2, 2, 5, x), x, ∞, 6))$Out$-\frac{16}{x^5}-\frac{5}{x^6}+(24(1-2\ln(-x)))/x^3+(6(2\ln(-x)-5))/x^2+(6(6\ln(-x)+7))/x^4$

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In$\operatorname{Normal}(\operatorname{Series}(\mathrm{Hypergeometric2F1}(3, 1, 2, x), x, ∞, 5))$Out$-\frac{1}{2x}+\frac{1}{2x^3}+\frac{3}{2x^5}+\frac{1}{x^4}$

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In$\operatorname{Normal}(\operatorname{Series}(\mathrm{Hypergeometric2F1}(2, 3, 4, x), x, ∞, 2))$Out$\frac{3}{x^2}$

About

Examples	12
Checked against	Wolfram Language 11 agree1 differSageMath 5 agree1 differ5 unevaluatedmpmath 4 agreeJulia 4 agree
Crosswalk	WikipediaMathWorldWikidataEncyclopedia of MathematicsBritannicaDLMFFungrimFungrim entry ×32Wolfram LanguageSageMathmpmathJulia

Implementation

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts