Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b8op)

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=-41/8, b>=a

For fixed z and a=-41/8, b=-35/8

http://functions.wolfram.com/07.23.03.b8op.01

Input Form

Hypergeometric2F1[-(41/8), -(35/8), 5, z] == (32768 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (4 (-62346240 + 1585283040 z - 23718746205 z^2 + 359139609675 z^3 + 36701999050463 z^4 + 143445435331255 z^5 + 148689217459785 z^6 + 46472508510945 z^7 + 3572654176245 z^8 + 20143021437 z^9) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (7793280 - 192498075 z + 2825530785 z^2 + 7863372902385 z^3 + 34418297134349 z^4 + 38562258575359 z^5 + 12894950721939 z^6 + 1061490837099 z^7 + 6714340479 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 5 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (-7793280 + 196881795 z - 2932931925 z^2 + 4788156344895 z^3 + 23273354349271 z^4 + 31175463930857 z^5 + 14337458779905 z^6 + 2194425771285 z^7 + 81664937277 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 2 (-62346240 + 1585283040 z - 23718746205 z^2 + 359139609675 z^3 + 36701999050463 z^4 + 143445435331255 z^5 + 148689217459785 z^6 + 46472508510945 z^7 + 3572654176245 z^8 + 20143021437 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (777213614249797725 Pi 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]