Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.aj6j)

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=3/4, b>=a

For fixed z and a=3/4, b=19/4

http://functions.wolfram.com/07.23.03.aj6j.01

Input Form

Hypergeometric2F1[3/4, 19/4, -(11/2), z] == (1/(1951488 Pi^(3/2))) (((1/(-1 + z)^11) (2 (-487872 + 5621616 z - 29983800 z^2 + 98906115 z^3 - 230221255 z^4 + 427694366 z^5 - 945070918 z^6 - 160429425 z^7 + 33597525 z^8 - 5357040 z^9 + 424320 z^10) EllipticE[(1/2) (1 - Sqrt[z])]) + (1/(-1 + z)^11) (2 (-487872 + 5621616 z - 29983800 z^2 + 98906115 z^3 - 230221255 z^4 + 427694366 z^5 - 945070918 z^6 - 160429425 z^7 + 33597525 z^8 - 5357040 z^9 + 424320 z^10) EllipticE[(1/2) (1 + Sqrt[z])]) - ((-487872 + 243936 Sqrt[z] + 5377680 z - 2587200 z^(3/2) - 27396600 z^2 + 12628770 z^(5/2) + 86277345 z^3 - 37963090 z^(7/2) - 192258165 z^4 + 80715140 z^(9/2) + 346979226 z^5 - 141017668 z^(11/2) - 804053250 z^6 - 138600150 z^(13/2) - 21829275 z^7 + 29868150 z^(15/2) + 3729375 z^8 - 5038800 z^(17/2) - 318240 z^9 + 424320 z^(19/2)) EllipticK[(1/2) (1 - Sqrt[z])])/ ((-1 + Sqrt[z])^11 (1 + Sqrt[z])^10) - ((487872 + 243936 Sqrt[z] - 5377680 z - 2587200 z^(3/2) + 27396600 z^2 + 12628770 z^(5/2) - 86277345 z^3 - 37963090 z^(7/2) + 192258165 z^4 + 80715140 z^(9/2) - 346979226 z^5 - 141017668 z^(11/2) + 804053250 z^6 - 138600150 z^(13/2) + 21829275 z^7 + 29868150 z^(15/2) - 3729375 z^8 - 5038800 z^(17/2) + 318240 z^9 + 424320 z^(19/2)) EllipticK[(1/2) (1 + Sqrt[z])])/ ((-1 + Sqrt[z])^10 (1 + Sqrt[z])^11)) Gamma[1/4]^2)

Standard Form

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MathML Form

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type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 424320 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 19 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 318240 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5038800 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 17 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3729375 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 29868150 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 15 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 21829275 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> 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<times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 37963090 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 86277345 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 12628770 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 27396600 </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'> 2587200 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5377680 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 243936 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 487872 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <cn type='integer'> 2 </cn> </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]