Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.ba8h)

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=13/8

http://functions.wolfram.com/07.23.03.ba8h.01

Input Form

Hypergeometric2F1[-(39/8), 13/8, 5, z] == (65536 2^(1/4) (8 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (7118592 - 47679736 z + 120487731 z^2 - 81363282 z^3 + 197396320 z^4 - 207297486 z^5 + 121913253 z^6 - 38964288 z^7 + 5296896 z^8) EllipticE[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 4 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (7118592 - 47679736 z + 120487731 z^2 - 81363282 z^3 + 197396320 z^4 - 207297486 z^5 + 121913253 z^6 - 38964288 z^7 + 5296896 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 4 Sqrt[1 - z] (7118592 - 47679736 z + 120487731 z^2 - 81363282 z^3 + 197396320 z^4 - 207297486 z^5 + 121913253 z^6 - 38964288 z^7 + 5296896 z^8) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - (28474368 - 201396832 z + 550550793 z^2 - 488179692 z^3 - 2546309282 z^4 + 4570145316 z^5 - 4189587039 z^6 + 2240326176 z^7 - 664038144 z^8 + 84750336 z^9) EllipticK[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (111997574714025 Pi (1 + Sqrt[1 - z])^(1/4) 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]