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 Hypergeometric2F1

 http://functions.wolfram.com/07.23.03.ah1b.01

 Input Form

 Hypergeometric2F1[-(3/4), 21/4, 3/2, z] == (1/(3315 Pi^(3/2) Sqrt[z])) (2 (-((2 (693 - 9393 z + 25116 z^2 - 24576 z^3 + 8192 z^4) EllipticE[(1/2) (1 - Sqrt[z])])/(-1 + z)^3) + (2 (693 - 9393 z + 25116 z^2 - 24576 z^3 + 8192 z^4) EllipticE[(1/2) (1 + Sqrt[z])])/(-1 + z)^3 + (1/((-1 + Sqrt[z])^3 (1 + Sqrt[z])^2)) ((693 - 2004 Sqrt[z] - 7389 z + 12252 z^(3/2) + 12864 z^2 - 18432 z^(5/2) - 6144 z^3 + 8192 z^(7/2)) EllipticK[(1/2) (1 - Sqrt[z])]) - (1/((-1 + Sqrt[z])^2 (1 + Sqrt[z])^3)) ((-693 - 2004 Sqrt[z] + 7389 z + 12252 z^(3/2) - 12864 z^2 - 18432 z^(5/2) + 6144 z^3 + 8192 z^(7/2)) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[3/4]^2)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["3", "4"]]], ",", FractionBox["21", "4"], ",", FractionBox["3", "2"], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["3315", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", SqrtBox["z"]]]], RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["693", "-", RowBox[List["9393", " ", "z"]], "+", RowBox[List["25116", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["24576", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["8192", " ", 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"]], ")"]], "3"]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["693", "-", RowBox[List["9393", " ", "z"]], "+", RowBox[List["25116", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["24576", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["8192", " ", 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"]], ")"]], "3"]], "+", RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["z"]]], ")"]], "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]], "2"]]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["693", "-", RowBox[List["2004", " ", SqrtBox["z"]]], "-", RowBox[List["7389", " ", "z"]], "+", RowBox[List["12252", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["12864", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["18432", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "-", RowBox[List["6144", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["8192", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]]]], ")"]], " ", 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"]]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]], "3"]]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "693"]], "-", RowBox[List["2004", " ", SqrtBox["z"]]], "+", RowBox[List["7389", " ", "z"]], "+", RowBox[List["12252", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["12864", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["18432", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["6144", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["8192", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]]]], ")"]], " ", 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 ( - 3 4 , 21 4 ; 3 2 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox["3", "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["3", "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 z ( 2 ( - 2 ( 8192 z 4 - 24576 z 3 + 25116 z 2 - 9393 z + 693 ) E ( 1 2 ( 1 - z ) ) ( z - 1 ) 3 + 2 ( 8192 z 4 - 24576 z 3 + 25116 z 2 - 9393 z + 693 ) E ( 1 2 ( z + 1 ) ) ( z - 1 ) 3 + 1 ( z - 1 ) 3 ( z + 1 ) 2 ( 8192 z 7 / 2 - 6144 z 3 - 18432 z 5 / 2 + 12864 z 2 + 12252 z 3 / 2 - 7389 z - 2004 z + 693 ) K ( 1 2 ( 1 - z ) ) - 1 ( z - 1 ) 2 ( z + 1 ) 3 ( 8192 z 7 / 2 + 6144 z 3 - 18432 z 5 / 2 - 12864 z 2 + 12252 z 3 / 2 + 7389 z - 2004 z - 693 ) K ( 1 2 ( z + 1 ) ) ) Γ ( 3 4 ) 2 ) HypergeometricPFQ -1 3 4 21 4 3 2 z 1 3315 3 2 z 1 2 -1 2 -1 2 8192 z 4 -1 24576 z 3 25116 z 2 -1 9393 z 693 EllipticE 1 2 1 -1 z 1 2 z -1 3 -1 2 8192 z 4 -1 24576 z 3 25116 z 2 -1 9393 z 693 EllipticE 1 2 z 1 2 1 z -1 3 -1 1 z 1 2 -1 3 z 1 2 1 2 -1 8192 z 7 2 -1 6144 z 3 -1 18432 z 5 2 12864 z 2 12252 z 3 2 -1 7389 z -1 2004 z 1 2 693 EllipticK 1 2 1 -1 z 1 2 -1 1 z 1 2 -1 2 z 1 2 1 3 -1 8192 z 7 2 6144 z 3 -1 18432 z 5 2 -1 12864 z 2 12252 z 3 2 7389 z -1 2004 z 1 2 -693 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[RowBox[List["-", FractionBox["3", "4"]]], ",", FractionBox["21", "4"], ",", FractionBox["3", "2"], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["693", "-", RowBox[List["9393", " ", "z"]], "+", RowBox[List["25116", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["24576", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["8192", " ", 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"]], ")"]], "3"]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["693", "-", RowBox[List["9393", " ", "z"]], "+", RowBox[List["25116", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["24576", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["8192", " ", 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"]], ")"]], "3"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["693", "-", RowBox[List["2004", " ", SqrtBox["z"]]], "-", RowBox[List["7389", " ", "z"]], "+", RowBox[List["12252", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["12864", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["18432", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "-", RowBox[List["6144", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["8192", " ", SuperscriptBox["z", RowBox[List["7", "/", "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"]]], ")"]], "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]], "2"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "693"]], "-", RowBox[List["2004", " ", SqrtBox["z"]]], "+", RowBox[List["7389", " ", "z"]], "+", RowBox[List["12252", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["12864", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["18432", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["6144", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["8192", " ", SuperscriptBox["z", RowBox[List["7", "/", "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"]]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]], "3"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["3", "4"], "]"]], "2"]]], RowBox[List["3315", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", SqrtBox["z"]]]]]]]]

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

 2007-05-02