Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b53p)

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=5/8

http://functions.wolfram.com/07.23.03.b53p.01

Input Form

Hypergeometric2F1[-(47/8), 5/8, 5, z] == (65536 2^(1/4) (-4 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (223049216 - 2404749360 z + 12291667245 z^2 - 43333409210 z^3 - 30510079578 z^4 + 22074407784 z^5 - 11917329435 z^6 + 4355662410 z^7 - 957052800 z^8 + 95344128 z^9) EllipticE[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (223049216 - 2404749360 z + 12291667245 z^2 - 43333409210 z^3 - 30510079578 z^4 + 22074407784 z^5 - 11917329435 z^6 + 4355662410 z^7 - 957052800 z^8 + 95344128 z^9) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 2 Sqrt[1 - z] (223049216 - 2404749360 z + 12291667245 z^2 - 43333409210 z^3 - 30510079578 z^4 + 22074407784 z^5 - 11917329435 z^6 + 4355662410 z^7 - 957052800 z^8 + 95344128 z^9) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (446098432 - 4976785632 z + 26341153995 z^2 - 95416708075 z^3 + 455229865134 z^4 - 320156142966 z^5 + 218949993207 z^6 - 111261253455 z^7 + 38432892960 z^8 - 8021908224 z^9 + 762753024 z^10) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (15926055124334355 Pi (1 + Sqrt[1 - z])^(1/4) z^4)

Standard Form

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MathML Form

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</annotation-xml> </semantics> </math>

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]