Gauss hypergeometric function 2F1: Integral transforms (formula 07.23.22.0001)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Integral transforms

Laplace transforms

http://functions.wolfram.com/07.23.22.0001.01

Input Form

LaplaceTransform[Hypergeometric2F1[a, b, c, -t], t, z] == ((Gamma[1 - a] Gamma[-a + b] Gamma[c])/(Gamma[b] Gamma[-a + c])) z^(a - 1) Hypergeometric1F1[1 + a - c, 1 + a - b, z] + ((Gamma[1 - b] Gamma[a - b] Gamma[c])/(Gamma[a] Gamma[-b + c])) z^(b - 1) Hypergeometric1F1[1 + b - c, 1 - a + b, z] + ((c - 1)/((a - 1) (b - 1))) HypergeometricPFQ[{1, 2 - c}, {2 - a, 2 - b}, z] /; Re[z] > 0

Standard Form

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MathML Form

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</mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> </annotation-xml> </semantics> </math>

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]