Gauss hypergeometric function 2F1: Integral representations (formula 07.23.07.0007)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Integral representations

Contour integral representations

http://functions.wolfram.com/07.23.07.0007.01

Input Form

Hypergeometric2F1[a, b, c, z] == (1/(2 Pi I)) (Gamma[c]/(Gamma[a] Gamma[b] Gamma[c - a] Gamma[c - b])) Integrate[(Gamma[s] Gamma[s + c - a - b] Gamma[a - s] Gamma[b - s])/ (1 - z)^s, {s, \[Gamma] - I Infinity, \[Gamma] + I Infinity}] /; Max[0, Re[a + b - c]] < \[Gamma] < Min[Re[a], Re[b]] && Abs[Arg[1 - z]] < Pi

Standard Form

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MathML Form

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Rule Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29

HypergeometricPFQ[{},{},z]
HypergeometricPFQ[{},{b},z]
HypergeometricPFQ[{a},{},z]
HypergeometricPFQ[{a},{b},z]
HypergeometricPFQ[{a₁},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅},{b₁,b₂,b₃,b₄},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅,a₆},{b₁,b₂,b₃,b₄,b₅},z]
HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]