Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.bbh5)

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=-37/8, b>=a

For fixed z and a=-37/8, b=-9/8

http://functions.wolfram.com/07.23.03.bbh5.01

Input Form

Hypergeometric2F1[-(37/8), -(9/8), 6, z] == (524288 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (457080832 - 6700876416 z + 49119715671 z^2 - 256521245162 z^3 + 1327877914545 z^4 + 31303276079460 z^5 + 21289523409385 z^6 + 34985211030 z^7 - 3390002385 z^8 + 193208400 z^9) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[1 - z] (457080832 - 6700876416 z + 49119715671 z^2 - 256521245162 z^3 + 1327877914545 z^4 + 31303276079460 z^5 + 21289523409385 z^6 + 34985211030 z^7 - 3390002385 z^8 + 193208400 z^9) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (457080832 - 6700876416 z + 49119715671 z^2 - 256521245162 z^3 + 1327877914545 z^4 + 31303276079460 z^5 + 21289523409385 z^6 + 34985211030 z^7 - 3390002385 z^8 + 193208400 z^9) EllipticK[ 1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - (457080832 - 6872281728 z + 51585675687 z^2 - 274278935918 z^3 + 1419385418997 z^4 + 5075960084580 z^5 - 4455570836255 z^6 - 1943864882190 z^7 + 146356168035 z^8 - 13930325640 z^9 + 772833600 z^10) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (6751051196550385725 Pi (1 + Sqrt[1 - z])^(1/4) (1 - z)^(1/4) z^5)

Standard Form

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MathML Form

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<sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <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> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 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<power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 274278935918 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 51585675687 </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'> 6872281728 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 457080832 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <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> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn 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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]