Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b4uc)

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=-47/8, b>=a

For fixed z and a=-47/8, b=-19/8

http://functions.wolfram.com/07.23.03.b4uc.01

Input Form

Hypergeometric2F1[-(47/8), -(19/8), 6, z] == (524288 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (271227846656 - 5349333585024 z + 55332889387971 z^2 - 436040373069379 z^3 + 3769845926111811 z^4 + 77155732052472549 z^5 + 117437608008152369 z^6 + 34581202390982799 z^7 + 147015982243449 z^8 - 9265279246641 z^9 + 369932959440 z^10) EllipticE[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (271227846656 - 5349333585024 z + 55332889387971 z^2 - 436040373069379 z^3 + 3769845926111811 z^4 + 77155732052472549 z^5 + 117437608008152369 z^6 + 34581202390982799 z^7 + 147015982243449 z^8 - 9265279246641 z^9 + 369932959440 z^10) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[1 - z] (271227846656 - 5349333585024 z + 55332889387971 z^2 - 436040373069379 z^3 + 3769845926111811 z^4 + 77155732052472549 z^5 + 117437608008152369 z^6 + 34581202390982799 z^7 + 147015982243449 z^8 - 9265279246641 z^9 + 369932959440 z^10) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - (271227846656 - 5451044027520 z + 57311078033235 z^2 - 456256465771405 z^3 + 3927969012676215 z^4 - 69280961392211457 z^5 - 237878645842493215 z^6 - 148432769127352935 z^7 - 13894288902796995 z^8 + 605665030906485 z^9 - 37770155158824 z^10 + 1479731837760 z^11) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/(38027888396142416877105 Pi (1 + Sqrt[1 - z])^(1/4) z^5)

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]