Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.9003)

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=-23/4, b>=a

For fixed z and a=-23/4, b=-21/4

http://functions.wolfram.com/07.23.03.9003.01

Input Form

Hypergeometric2F1[-(23/4), -(21/4), 4, -z] == (1/(22563055139625 Pi z^3)) (256 (1 + z)^(1/4) (-2 (3090464 + 110966973 z + 3113159595 z^2 - 311879449215 z^3 + 1843593377175 z^4 - 3036960518697 z^5 + 1659945989121 z^6 - 281343839445 z^7 + 10097123805 z^8) EllipticE[1/2 - 1/(2 Sqrt[1 + z])] + (3090464 (1 + Sqrt[1 + z]) + 10140585 z^2 (307 + 299 Sqrt[1 + z]) + 289731 z (383 + 375 Sqrt[1 + z]) + 45 z^8 (224380529 + 7081945 Sqrt[1 + z]) - 45 z^7 (6252085321 + 606627153 Sqrt[1 + z]) + 225 z^4 (8193748343 + 2693379023 Sqrt[1 + z]) - 45 z^3 (6930654427 + 3063711347 Sqrt[1 + z]) - 63 z^5 (48205722519 + 11678773199 Sqrt[1 + z]) + 21 z^6 (79045047101 + 13167762245 Sqrt[1 + z])) EllipticK[1/2 - 1/(2 Sqrt[1 + z])]))

Standard Form

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MathML Form

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</apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </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["23", "4"]]], ",", RowBox[List["-", FractionBox["21", "4"]]], ",", "4", ",", RowBox[List["-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["256", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["3090464", "+", RowBox[List["110966973", " ", "z"]], "+", RowBox[List["3113159595", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["311879449215", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["1843593377175", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["3036960518697", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["1659945989121", " ", SuperscriptBox["z", "6"]]], "-", RowBox[List["281343839445", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["10097123805", " ", SuperscriptBox["z", "8"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox["1", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3090464", " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "+", "z"]]]]], ")"]]]], "+", RowBox[List["10140585", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["307", "+", RowBox[List["299", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]], "+", RowBox[List["289731", " ", "z", " ", RowBox[List["(", RowBox[List["383", "+", RowBox[List["375", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]], "+", RowBox[List["45", " ", SuperscriptBox["z", "8"], " ", RowBox[List["(", RowBox[List["224380529", "+", RowBox[List["7081945", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]], "-", RowBox[List["45", " ", SuperscriptBox["z", "7"], " ", RowBox[List["(", RowBox[List["6252085321", "+", RowBox[List["606627153", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]], "+", RowBox[List["225", " ", SuperscriptBox["z", "4"], " ", RowBox[List["(", RowBox[List["8193748343", "+", RowBox[List["2693379023", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]], "-", RowBox[List["45", " ", SuperscriptBox["z", "3"], " ", RowBox[List["(", RowBox[List["6930654427", "+", RowBox[List["3063711347", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]], "-", RowBox[List["63", " ", SuperscriptBox["z", "5"], " ", RowBox[List["(", RowBox[List["48205722519", "+", RowBox[List["11678773199", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]], "+", RowBox[List["21", " ", SuperscriptBox["z", "6"], " ", RowBox[List["(", RowBox[List["79045047101", "+", RowBox[List["13167762245", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox["1", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List["22563055139625", " ", "\[Pi]", " ", 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]