Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.acw2)

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=-13/4, b>=a

For fixed z and a=-13/4, b=23/4

http://functions.wolfram.com/07.23.03.acw2.01

Input Form

Hypergeometric2F1[-(13/4), 23/4, -(7/2), z] == (1/(9831360 Pi^(3/2))) (((1/(-1 + z)^6) (2 (2457840 - 1316700 z - 2052589 z^2 - 5914909 z^3 - 49521087 z^4 + 337180085 z^5 - 684688000 z^6 + 658225152 z^7 - 311492608 z^8 + 58720256 z^9) EllipticE[(1/2) (1 - Sqrt[z])]) + (1/(-1 + z)^6) (2 (2457840 - 1316700 z - 2052589 z^2 - 5914909 z^3 - 49521087 z^4 + 337180085 z^5 - 684688000 z^6 + 658225152 z^7 - 311492608 z^8 + 58720256 z^9) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/((-1 + Sqrt[z])^6 (1 + Sqrt[z])^5)) ((-2457840 + 1228920 Sqrt[z] + 87780 z + 468160 z^(3/2) + 1584429 z^2 - 554477 z^(5/2) + 6469386 z^3 - 3432198 z^(7/2) + 52953285 z^4 - 107386565 z^(9/2) - 229793520 z^5 + 365061760 z^(11/2) + 319626240 z^6 - 464517120 z^(13/2) - 193708032 z^7 + 267452416 z^(15/2) + 44040192 z^8 - 58720256 z^(17/2)) EllipticK[(1/2) (1 - Sqrt[z])]) - (1/((-1 + Sqrt[z])^5 (1 + Sqrt[z])^6)) ((-2457840 - 1228920 Sqrt[z] + 87780 z - 468160 z^(3/2) + 1584429 z^2 + 554477 z^(5/2) + 6469386 z^3 + 3432198 z^(7/2) + 52953285 z^4 + 107386565 z^(9/2) - 229793520 z^5 - 365061760 z^(11/2) + 319626240 z^6 + 464517120 z^(13/2) - 193708032 z^7 - 267452416 z^(15/2) + 44040192 z^8 + 58720256 z^(17/2)) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[1/4]^2)

Standard Form

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MathML Form

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Rule Form

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RowBox[List["58720256", " ", SuperscriptBox["z", RowBox[List["17", "/", "2"]]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["z"]]], ")"]], "5"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]], "6"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["1", "4"], "]"]], "2"]]], RowBox[List["9831360", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]]]]]]

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]