Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a97x)

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=-19/4, b>=a

For fixed z and a=-19/4, b=-7/4

http://functions.wolfram.com/07.23.03.a97x.01

Input Form

Hypergeometric2F1[-(19/4), -(7/4), 5, -z] == (4096 Sqrt[2] (Sqrt[1 + z] (-384 - 6224 z - 55811 z^2 - 469371 z^3 + 5332610 z^4 - 4231846 z^5 + 213969 z^6 + 14881 z^7 + 736 z^8) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + (-384 - 6608 z - 62035 z^2 - 525182 z^3 + 4863239 z^4 + 1100764 z^5 - 4017877 z^6 + 228850 z^7 + 15617 z^8 + 736 z^9) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - 4 (-96 - 1628 z - 15113 z^2 - 127701 z^3 - 730780 z^4 + 3112706 z^5 - 1361577 z^6 + 943 z^7 + 46 z^8) EllipticK[(-1 + Sqrt[1 + z])/ (1 + Sqrt[1 + z])] - Sqrt[1 + z] (-384 - 6224 z - 55811 z^2 - 469371 z^3 + 5332610 z^4 - 4231846 z^5 + 213969 z^6 + 14881 z^7 + 736 z^8) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])]))/ (16194422475 Pi z^4 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 213969 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4231846 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 5332610 </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'> 469371 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 55811 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6224 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 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</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]