Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b9n5)

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=-39/8, b>=a

For fixed z and a=-39/8, b=-37/8

http://functions.wolfram.com/07.23.03.b9n5.01

Input Form

Hypergeometric2F1[-(39/8), -(37/8), 5, z] == (65536 2^(1/4) (8 Sqrt[1 - z] (-110699264 + 2764022248 z - 40902945629 z^2 + 621701768870 z^3 + 18909089871485 z^4 + 53122185628700 z^5 + 39589749778605 z^6 + 8180083432950 z^7 + 324572249235 z^8) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + 4 Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (-110699264 + 2764022248 z - 40902945629 z^2 + 621701768870 z^3 + 18909089871485 z^4 + 53122185628700 z^5 + 39589749778605 z^6 + 8180083432950 z^7 + 324572249235 z^8) EllipticE[ 2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - 4 Sqrt[1 - z] (-110699264 + 2764022248 z - 40902945629 z^2 + 621701768870 z^3 + 18909089871485 z^4 + 53122185628700 z^5 + 39589749778605 z^6 + 8180083432950 z^7 + 324572249235 z^8) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + (442797056 - 11332837152 z + 170476434141 z^2 - 2587952041675 z^3 + 96093611932065 z^4 + 594513116398665 z^5 + 841998490435895 z^6 + 358803156339615 z^7 + 41640282158475 z^8 + 725838098115 z^9) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/ (5576850641068782075 Pi Sqrt[Sqrt[2] + Sqrt[1 - Sqrt[1 - z]]] z^4)

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]