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

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=-33/8, b>=a

For fixed z and a=-33/8, b=45/8

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

Input Form

Hypergeometric2F1[-(33/8), 45/8, 5, z] == (1/(2038647453017175 Pi z^4)) (32768 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (8 (-4308480 - 19163760 z - 89628165 z^2 - 660728970 z^3 + 42456019327 z^4 - 173954534352 z^5 + 274054145904 z^6 - 192267694080 z^7 + 50470269696 z^8) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 5 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (-1077120 - 4967655 z - 23275890 z^2 + 12273639345 z^3 - 53962707272 z^4 + 87976553328 z^5 - 63087837120 z^6 + 16823423232 z^7) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (1077120 + 5573535 z + 26532495 z^2 + 20923084050 z^3 - 97318774288 z^4 + 165278504160 z^5 - 122570654976 z^6 + 33646846464 z^7) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 4 (-4308480 - 19163760 z - 89628165 z^2 - 660728970 z^3 + 42456019327 z^4 - 173954534352 z^5 + 274054145904 z^6 - 192267694080 z^7 + 50470269696 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))

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]