Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.aa9g)

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=-17/4, b>=a

For fixed z and a=-17/4, b=-13/4

http://functions.wolfram.com/07.23.03.aa9g.01

Input Form

Hypergeometric2F1[-(17/4), -(13/4), 3/2, z] == (1/(10138590 Pi^(3/2))) ((2 (2470005 + 21329580 z + 28422462 z^2 + 6453164 z^3 + 45045 z^4) EllipticE[(1/2) (1 - Sqrt[z])] + 2 (2470005 + 21329580 z + 28422462 z^2 + 6453164 z^3 + 45045 z^4) EllipticE[(1/2) (1 + Sqrt[z])] - (1/Sqrt[z]) ((129285 + 2470005 Sqrt[z] + 5824620 z + 21329580 z^(3/2) + 26017950 z^2 + 28422462 z^(5/2) + 23099756 z^3 + 6453164 z^(7/2) + 3648645 z^4 + 45045 z^(9/2)) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/Sqrt[z]) ((129285 - 2470005 Sqrt[z] + 5824620 z - 21329580 z^(3/2) + 26017950 z^2 - 28422462 z^(5/2) + 23099756 z^3 - 6453164 z^(7/2) + 3648645 z^4 - 45045 z^(9/2)) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[1/4]^2)

Standard Form

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MathML Form

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</cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["17", "4"]]], ",", RowBox[List["-", FractionBox["13", "4"]]], ",", FractionBox["3", "2"], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["2470005", "+", RowBox[List["21329580", " ", "z"]], "+", RowBox[List["28422462", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["6453164", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["45045", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["2470005", "+", RowBox[List["21329580", " ", "z"]], "+", RowBox[List["28422462", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["6453164", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["45045", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["129285", "+", RowBox[List["2470005", " ", SqrtBox["z"]]], "+", RowBox[List["5824620", " ", "z"]], "+", RowBox[List["21329580", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["26017950", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["28422462", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["23099756", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["6453164", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "+", RowBox[List["3648645", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["45045", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], SqrtBox["z"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["129285", "-", RowBox[List["2470005", " ", SqrtBox["z"]]], "+", RowBox[List["5824620", " ", "z"]], "-", RowBox[List["21329580", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["26017950", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["28422462", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["23099756", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["6453164", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "+", RowBox[List["3648645", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["45045", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], SqrtBox["z"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["1", "4"], "]"]], "2"]]], RowBox[List["10138590", " ", 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]