Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b7cf)

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=-45/8, b>=a

For fixed z and a=-45/8, b=47/8

http://functions.wolfram.com/07.23.03.b7cf.01

Input Form

Hypergeometric2F1[-(45/8), 47/8, 6, z] == (524288 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (105480192 + 253399680 z + 760311705 z^2 + 3116201235 z^3 + 22782552975 z^4 - 502684717675 z^5 + 1886484827920 z^6 - 3185239875840 z^7 + 2814459985920 z^8 - 1274850508800 z^9 + 234712203264 z^10) EllipticE[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[1 - z] (105480192 + 253399680 z + 760311705 z^2 + 3116201235 z^3 + 22782552975 z^4 - 502684717675 z^5 + 1886484827920 z^6 - 3185239875840 z^7 + 2814459985920 z^8 - 1274850508800 z^9 + 234712203264 z^10) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (105480192 + 253399680 z + 760311705 z^2 + 3116201235 z^3 + 22782552975 z^4 - 502684717675 z^5 + 1886484827920 z^6 - 3185239875840 z^7 + 2814459985920 z^8 - 1274850508800 z^9 + 234712203264 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (105480192 + 213844608 z + 654470985 z^2 + 2799355065 z^3 + 21518451675 z^4 + 651658113995 z^5 - 4716256889960 z^6 + 12690666077760 z^7 - 17642409861120 z^8 + 13601469235200 z^9 - 5549267091456 z^10 + 938848813056 z^11) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (304941368586659805 Pi (1 + Sqrt[1 - z])^(1/4) (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]