Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.0180)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

Specialized values

For fixed z and with symbolical integers in parameters

For fixed z and a=1, b=m/n, c=m/n+1

http://functions.wolfram.com/07.23.03.0180.01

Input Form

Hypergeometric2F1[1, m/n, m/n + 1, z] == ((-(m/n)) (Log[1 - z^(1/n)] + (((-1)^m (1 + (-1)^n))/2) Log[1 + z^(1/n)] + Sum[Cos[(2 Pi k m)/n] Log[1 - 2 z^(1/n) Cos[(2 Pi k)/n] + z^(2/n)] - 2 Sin[(2 Pi k m)/n] ArcTan[(z^(1/n) Sin[(2 Pi k)/n])/ (1 - z^(1/n) Cos[(2 Pi k)/n])], {k, 1, Floor[(n - 1)/2]}]))/ z^(m/n) /; Element[m, Integers] && m > 0 && Element[n, Integers] && n > 0 && n > m

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]