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

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=-37/8, b>=a

For fixed z and a=-37/8, b=-25/8

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

Input Form

Hypergeometric2F1[-(37/8), -(25/8), 5, z] == (65536 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-1214120960 + 24775655840 z - 288482965485 z^2 + 3264640838290 z^3 + 203938093747913 z^4 + 497822382003660 z^5 + 274186326982645 z^6 + 31339402776290 z^7 + 3900394575 z^8) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[1 - z] (-1214120960 + 24775655840 z - 288482965485 z^2 + 3264640838290 z^3 + 203938093747913 z^4 + 497822382003660 z^5 + 274186326982645 z^6 + 31339402776290 z^7 + 3900394575 z^8) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-1214120960 + 24775655840 z - 288482965485 z^2 + 3264640838290 z^3 + 203938093747913 z^4 + 497822382003660 z^5 + 274186326982645 z^6 + 31339402776290 z^7 + 3900394575 z^8) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - 4 (-303530240 + 6307737800 z - 74412335400 z^2 + 842586903925 z^3 + 12557393193512 z^4 + 7553071040613 z^5 - 14021150704400 z^6 - 6555361134385 z^7 - 312031566000 z^8 + 3900394575 z^9) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (4997531405238597225 Pi (1 + Sqrt[1 - z])^(1/4) (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]