Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.biph)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 8 and fixed z and a<0

For fixed z and a=-25/8, b>=a

For fixed z and a=-25/8, b=-21/8

http://functions.wolfram.com/07.23.03.biph.01

Input Form

Hypergeometric2F1[-(25/8), -(21/8), 4, z] == (2048 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (141440 - 2425475 z + 28691325 z^2 + 1505178682 z^3 + 2539411490 z^4 + 716654505 z^5 + 19040385 z^6) EllipticE[ 1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[1 - z] (141440 - 2425475 z + 28691325 z^2 + 1505178682 z^3 + 2539411490 z^4 + 716654505 z^5 + 19040385 z^6) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (141440 - 2425475 z + 28691325 z^2 + 1505178682 z^3 + 2539411490 z^4 + 716654505 z^5 + 19040385 z^6) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (-141440 + 2478515 z - 29586375 z^2 - 343204882 z^3 + 11407522 z^4 + 312485775 z^5 + 46560885 z^6) EllipticK[ 1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (1179090487575 Pi (1 + Sqrt[1 - z])^(1/4) (1 - z)^(1/4) z^3)

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)

2007-05-02

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]