Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b7ii)

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=-43/8, b>=a

For fixed z and a=-43/8, b=-23/8

http://functions.wolfram.com/07.23.03.b7ii.01

Input Form

Hypergeometric2F1[-(43/8), -(23/8), 5, z] == (65536 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (-1212244992 + 25646558112 z - 312076136295 z^2 + 3739574548710 z^3 + 107104157731355 z^4 + 221696371075892 z^5 + 100124630251583 z^6 + 6854637852790 z^7 - 186919163795 z^8 + 5611081840 z^9) EllipticE[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 2 (-606122496 + 13050574992 z - 160784646561 z^2 + 1927019680740 z^3 - 41978476346915 z^4 - 195137194763114 z^5 - 171766655195207 z^6 - 31923979716664 z^7 - 23496405205 z^8 + 701385230 z^9) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (-1212244992 + 25646558112 z - 312076136295 z^2 + 3739574548710 z^3 + 107104157731355 z^4 + 221696371075892 z^5 + 100124630251583 z^6 + 6854637852790 z^7 - 186919163795 z^8 + 5611081840 z^9) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[1 - z] (-1212244992 + 25646558112 z - 312076136295 z^2 + 3739574548710 z^3 + 107104157731355 z^4 + 221696371075892 z^5 + 100124630251583 z^6 + 6854637852790 z^7 - 186919163795 z^8 + 5611081840 z^9) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/(6215742787247618625 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]