Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b9bf)

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=-41/8, b>=a

For fixed z and a=-41/8, b=27/8

http://functions.wolfram.com/07.23.03.b9bf.01

Input Form

Hypergeometric2F1[-(41/8), 27/8, 6, z] == (524288 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (114196480 - 460800640 z + 413861175 z^2 + 247365300 z^3 + 580357050 z^4 - 18764227404 z^5 + 48807159687 z^6 - 58697664480 z^7 + 38434556160 z^8 - 13349539840 z^9 + 1938227200 z^10) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (114196480 - 503624320 z + 574951815 z^2 + 133196700 z^3 + 466188450 z^4 - 6091512804 z^5 + 14264967951 z^6 - 16205924592 z^7 + 10191242880 z^8 - 3428239360 z^9 + 484556800 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[1 - z] (114196480 - 460800640 z + 413861175 z^2 + 247365300 z^3 + 580357050 z^4 - 18764227404 z^5 + 48807159687 z^6 - 58697664480 z^7 + 38434556160 z^8 - 13349539840 z^9 + 1938227200 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (114196480 - 460800640 z + 413861175 z^2 + 247365300 z^3 + 580357050 z^4 - 18764227404 z^5 + 48807159687 z^6 - 58697664480 z^7 + 38434556160 z^8 - 13349539840 z^9 + 1938227200 z^10) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (3388155526030275 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]