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 Hypergeometric2F1

 http://functions.wolfram.com/07.23.03.ak38.01

 Input Form

 Hypergeometric2F1[5/4, 21/4, 1/2, z] == (1/(3315 Pi^(3/2))) (((8 Sqrt[z] (-7815 - 3750 z + 1857 z^2 - 628 z^3 + 96 z^4) EllipticE[(1/2) (1 - Sqrt[z])])/(-1 + z)^6 - (8 Sqrt[z] (-7815 - 3750 z + 1857 z^2 - 628 z^3 + 96 z^4) EllipticE[(1/2) (1 + Sqrt[z])])/(-1 + z)^6 + (1/((-1 + Sqrt[z])^6 (1 + Sqrt[z])^5)) ((3315 + 27945 Sqrt[z] + 10890 z + 4110 z^(3/2) - 5805 z^2 - 1623 z^(5/2) + 2224 z^3 + 288 z^(7/2) - 384 z^4) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z])^5 (1 + Sqrt[z])^6)) ((-3315 + 27945 Sqrt[z] - 10890 z + 4110 z^(3/2) + 5805 z^2 - 1623 z^(5/2) - 2224 z^3 + 288 z^(7/2) + 384 z^4) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[3/4]^2)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["5", "4"], ",", FractionBox["21", "4"], ",", FractionBox["1", "2"], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["3315", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["8", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "7815"]], "-", RowBox[List["3750", " ", "z"]], "+", RowBox[List["1857", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["628", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["96", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "6"]], "-", FractionBox[RowBox[List["8", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "7815"]], "-", RowBox[List["3750", " ", "z"]], "+", RowBox[List["1857", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["628", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["96", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "6"]], "+", RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["z"]]], ")"]], "6"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]], "5"]]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["3315", "+", RowBox[List["27945", " ", SqrtBox["z"]]], "+", RowBox[List["10890", " ", "z"]], "+", RowBox[List["4110", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["5805", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["1623", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["2224", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["288", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "-", RowBox[List["384", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["z"]]], ")"]], "5"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]], "6"]]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3315"]], "+", RowBox[List["27945", " ", SqrtBox["z"]]], "-", RowBox[List["10890", " ", "z"]], "+", RowBox[List["4110", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["5805", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["1623", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "-", RowBox[List["2224", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["288", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "+", RowBox[List["384", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["3", "4"], "]"]], "2"]]], ")"]]]]]]]]

 MathML Form

 2 F 1 ( 5 4 , 21 4 ; 1 2 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["5", "4"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["21", "4"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] 1 3315 π 3 / 2 ( ( 8 z ( 96 z 4 - 628 z 3 + 1857 z 2 - 3750 z - 7815 ) E ( 1 2 ( 1 - z ) ) ( z - 1 ) 6 - 8 z ( 96 z 4 - 628 z 3 + 1857 z 2 - 3750 z - 7815 ) E ( 1 2 ( z + 1 ) ) ( z - 1 ) 6 + 1 ( z - 1 ) 6 ( z + 1 ) 5 ( ( - 384 z 4 + 288 z 7 / 2 + 2224 z 3 - 1623 z 5 / 2 - 5805 z 2 + 4110 z 3 / 2 + 10890 z + 27945 z + 3315 ) K ( 1 2 ( 1 - z ) ) ) + 1 ( z - 1 ) 5 ( z + 1 ) 6 ( 384 z 4 + 288 z 7 / 2 - 2224 z 3 - 1623 z 5 / 2 + 5805 z 2 + 4110 z 3 / 2 - 10890 z + 27945 z - 3315 ) K ( 1 2 ( z + 1 ) ) ) Γ ( 3 4 ) 2 ) HypergeometricPFQ 5 4 21 4 1 2 z 1 3315 3 2 -1 8 z 1 2 96 z 4 -1 628 z 3 1857 z 2 -1 3750 z -7815 EllipticE 1 2 1 -1 z 1 2 z -1 6 -1 -1 8 z 1 2 96 z 4 -1 628 z 3 1857 z 2 -1 3750 z -7815 EllipticE 1 2 z 1 2 1 z -1 6 -1 1 z 1 2 -1 6 z 1 2 1 5 -1 -384 z 4 288 z 7 2 2224 z 3 -1 1623 z 5 2 -1 5805 z 2 4110 z 3 2 10890 z 27945 z 1 2 3315 EllipticK 1 2 1 -1 z 1 2 1 z 1 2 -1 5 z 1 2 1 6 -1 384 z 4 288 z 7 2 -1 2224 z 3 -1 1623 z 5 2 5805 z 2 4110 z 3 2 -1 10890 z 27945 z 1 2 -3315 EllipticK 1 2 z 1 2 1 Gamma 3 4 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["5", "4"], ",", FractionBox["21", "4"], ",", FractionBox["1", "2"], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["8", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "7815"]], "-", RowBox[List["3750", " ", "z"]], "+", RowBox[List["1857", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["628", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["96", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "6"]], "-", FractionBox[RowBox[List["8", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "7815"]], "-", RowBox[List["3750", " ", "z"]], "+", RowBox[List["1857", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["628", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["96", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "6"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["3315", "+", RowBox[List["27945", " ", SqrtBox["z"]]], "+", RowBox[List["10890", " ", "z"]], "+", RowBox[List["4110", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["5805", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["1623", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["2224", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["288", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "-", RowBox[List["384", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", 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"]]], ")"]], "6"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]], "5"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3315"]], "+", RowBox[List["27945", " ", SqrtBox["z"]]], "-", RowBox[List["10890", " ", "z"]], "+", RowBox[List["4110", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["5805", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["1623", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "-", RowBox[List["2224", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["288", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "+", RowBox[List["384", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", 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["3", "4"], "]"]], "2"]]], RowBox[List["3315", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02