Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b6k3)

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=-45/8, b>=a

For fixed z and a=-45/8, b=-1/8

http://functions.wolfram.com/07.23.03.b6k3.01

Input Form

Hypergeometric2F1[-(45/8), -(1/8), 4, z] == (2048 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (12498304 - 167457745 z + 1351867335 z^2 + 32079476270 z^3 + 533659750 z^4 - 220945005 z^5 + 70799475 z^6 - 14422800 z^7 + 1370880 z^8) EllipticE[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[1 - z] (12498304 - 167457745 z + 1351867335 z^2 + 32079476270 z^3 + 533659750 z^4 - 220945005 z^5 + 70799475 z^6 - 14422800 z^7 + 1370880 z^8) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (12498304 - 167457745 z + 1351867335 z^2 + 32079476270 z^3 + 533659750 z^4 - 220945005 z^5 + 70799475 z^6 - 14422800 z^7 + 1370880 z^8) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - (12498304 - 172144609 z + 1413382425 z^2 + 4467988730 z^3 - 7494367550 z^4 + 2530851795 z^5 - 1013632395 z^6 + 310258500 z^7 - 60318720 z^8 + 5483520 z^9) EllipticK[ 1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (27771932076525 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]