Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b7df)

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=-43/8, b>=a

For fixed z and a=-43/8, b=-41/8

http://functions.wolfram.com/07.23.03.b7df.01

Input Form

Hypergeometric2F1[-(43/8), -(41/8), 2, z] == (1/(207179782425 Pi z)) (8 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (4 (82905075 + 44168560894 z + 525180057037 z^2 + 1409169604580 z^3 + 1104614292573 z^4 + 242687081406 z^5 + 9995241795 z^6) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 312 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (82905075 + 1127712233 z + 3314754782 z^2 + 2803138162 z^3 + 661083423 z^4 + 29391285 z^5) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 5 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (5189857695 + 74029678838 z + 241507466465 z^2 + 249909043828 z^3 + 87258642801 z^4 + 9119826870 z^5 + 165032175 z^6) EllipticK[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 2 (82905075 + 44168560894 z + 525180057037 z^2 + 1409169604580 z^3 + 1104614292573 z^4 + 242687081406 z^5 + 9995241795 z^6) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))

Standard Form

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MathML Form

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type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 165032175 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 9119826870 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 87258642801 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 249909043828 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 241507466465 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 74029678838 </cn> <ci> z </ci> </apply> <cn type='integer'> 5189857695 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='rational'> 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5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1104614292573 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1409169604580 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 525180057037 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 44168560894 </cn> <ci> z </ci> </apply> <cn type='integer'> 82905075 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </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]