Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b6h2)

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=-9/8

http://functions.wolfram.com/07.23.03.b6h2.01

Input Form

Hypergeometric2F1[-(45/8), -(9/8), 5, z] == (65536 2^(1/4) (-8 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (10712832 - 175311032 z + 1553374589 z^2 - 12327916965 z^3 - 442042314900 z^4 - 441519872970 z^5 - 1339463235 z^6 + 219927895 z^7 - 27984550 z^8 + 1840080 z^9) EllipticE[ 1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + 4 Sqrt[1 - z] (10712832 - 175311032 z + 1553374589 z^2 - 12327916965 z^3 - 442042314900 z^4 - 441519872970 z^5 - 1339463235 z^6 + 219927895 z^7 - 27984550 z^8 + 1840080 z^9) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + 4 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (10712832 - 175311032 z + 1553374589 z^2 - 12327916965 z^3 - 442042314900 z^4 - 441519872970 z^5 - 1339463235 z^6 + 219927895 z^7 - 27984550 z^8 + 1840080 z^9) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (42851328 - 717313376 z + 6472070969 z^2 - 51572159106 z^3 - 348742244685 z^4 + 215665389060 z^5 + 198625020495 z^6 - 23071459730 z^7 + 3730263845 z^8 - 461860080 z^9 + 29441280 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (45925518343880175 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]