Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.bcz7)

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=-35/8, b>=a

For fixed z and a=-35/8, b=23/8

http://functions.wolfram.com/07.23.03.bcz7.01

Input Form

Hypergeometric2F1[-(35/8), 23/8, 3, z] == (1/(433128465 Pi z^2)) (128 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (4 (-15048 - 84645 z + 1905978 z^2 - 5939749 z^3 + 7696104 z^4 - 4614480 z^5 + 1060800 z^6) EllipticE[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (1881 + 1139886 z - 4447455 z^2 + 6543368 z^3 - 4296240 z^4 + 1060800 z^5) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 5 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1881 - 665874 z + 2971209 z^2 - 5406544 z^3 + 4964544 z^4 - 2291328 z^5 + 424320 z^6) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 2 (-15048 - 84645 z + 1905978 z^2 - 5939749 z^3 + 7696104 z^4 - 4614480 z^5 + 1060800 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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</apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4614480 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 7696104 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5939749 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1905978 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 84645 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -15048 </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]