Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a8ge)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 4 and fixed z

For fixed z and a=-21/4, b>=a

For fixed z and a=-21/4, b=-1/4

http://functions.wolfram.com/07.23.03.a8ge.01

Input Form

Hypergeometric2F1[-(21/4), -(1/4), 7/2, z] == (1/(309806343 Pi^(3/2) z^(5/2))) (2 (2 Sqrt[z] (-19890 + 275145 z + 35488944 z^2 + 8575182 z^3 - 3228918 z^4 + 1054977 z^5 - 224928 z^6 + 22528 z^7) EllipticE[(1/2) (1 - Sqrt[z])] + 2 Sqrt[z] (-19890 + 275145 z + 35488944 z^2 + 8575182 z^3 - 3228918 z^4 + 1054977 z^5 - 224928 z^6 + 22528 z^7) EllipticE[(1/2) (1 + Sqrt[z])] - (39780 - 19890 Sqrt[z] - 580125 z + 275145 z^(3/2) + 6881940 z^2 + 35488944 z^(5/2) + 36142722 z^3 + 8575182 z^(7/2) - 741048 z^4 - 3228918 z^(9/2) + 248787 z^5 + 1054977 z^(11/2) - 54648 z^6 - 224928 z^(13/2) + 5632 z^7 + 22528 z^(15/2)) EllipticK[(1/2) (1 - Sqrt[z])] - (-39780 - 19890 Sqrt[z] + 580125 z + 275145 z^(3/2) - 6881940 z^2 + 35488944 z^(5/2) - 36142722 z^3 + 8575182 z^(7/2) + 741048 z^4 - 3228918 z^(9/2) - 248787 z^5 + 1054977 z^(11/2) + 54648 z^6 - 224928 z^(13/2) - 5632 z^7 + 22528 z^(15/2)) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[1/4]^2)

Standard Form

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MathML Form

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</apply> </apply> <apply> <times /> <cn type='integer'> 35488944 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6881940 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 275145 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 580125 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 19890 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 39780 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> 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</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]