Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a7yj)

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=-11/2, b>=a

For fixed z and a=-11/2, b=-5/4

http://functions.wolfram.com/07.23.03.a7yj.01

Input Form

Hypergeometric2F1[-(11/2), -(5/4), 4, z] == (2 (157696 - 2828672 z + 34367872 z^2 + 1141953728 z^3 + 1365919136 z^4 + 27335048 z^5 - 4529616 z^6 + 611286 z^7 - 43095 z^8) EllipticE[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])] + 2 Sqrt[1 - z] (157696 - 2828672 z + 34367872 z^2 + 1141953728 z^3 + 1365919136 z^4 + 27335048 z^5 - 4529616 z^6 + 611286 z^7 - 43095 z^8) EllipticE[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])] + (-157696 + 2828672 z - 34367872 z^2 - 1141953728 z^3 - 1365919136 z^4 - 27335048 z^5 + 4529616 z^6 - 611286 z^7 + 43095 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])] + (1 - z)^(1/4) (-157696 + 2828672 z - 34367872 z^2 - 1141953728 z^3 - 1365919136 z^4 - 27335048 z^5 + 4529616 z^6 - 611286 z^7 + 43095 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])] + Sqrt[1 - z] (-157696 + 2828672 z - 34367872 z^2 - 1141953728 z^3 - 1365919136 z^4 - 27335048 z^5 + 4529616 z^6 - 611286 z^7 + 43095 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])] - (1 - z)^(3/4) (157696 - 2749824 z + 33017600 z^2 + 481108928 z^3 + 363324000 z^4 - 24078392 z^5 + 4068168 z^6 - 576810 z^7 + 43095 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])])/ (169234065 Sqrt[2] Pi Sqrt[1 + Sqrt[1 - z]] z^3)

Standard Form

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MathML Form

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Rule Form

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RowBox[List["4068168", " ", SuperscriptBox["z", "6"]]], "-", RowBox[List["576810", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["43095", " ", SuperscriptBox["z", "8"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]], "]"]]]]]], RowBox[List["169234065", " ", SqrtBox["2"], " ", "\[Pi]", " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SuperscriptBox["z", "3"]]]]]]]]

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]