Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.9989)

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=17/4

http://functions.wolfram.com/07.23.03.9989.01

Input Form

Hypergeometric2F1[-(23/4), 17/4, 9/2, z] == (1/(22889262825 Pi^(3/2) z^(7/2))) (16 (8 (1009470 + 2927463 z + 13897037 z^2 + 181098918 z^3 - 1897593093 z^4 + 5827845485 z^5 - 8751926784 z^6 + 7117953024 z^7 - 3024093184 z^8 + 528482304 z^9) EllipticE[(1/2) (1 - Sqrt[z])] - 8 (1009470 + 2927463 z + 13897037 z^2 + 181098918 z^3 - 1897593093 z^4 + 5827845485 z^5 - 8751926784 z^6 + 7117953024 z^7 - 3024093184 z^8 + 528482304 z^9) EllipticE[(1/2) (1 + Sqrt[z])] - (4037880 + 2018940 Sqrt[z] + 11709852 z + 6023171 z^(3/2) + 55588148 z^2 + 28366107 z^(5/2) + 724395672 z^3 - 1065766383 z^(7/2) - 7590372372 z^4 + 4184013845 z^(9/2) + 23311381940 z^5 - 7157215680 z^(11/2) - 35007707136 z^6 + 6347937792 z^(13/2) + 28471812096 z^7 - 2875457536 z^(15/2) - 12096372736 z^8 + 528482304 z^(17/2) + 2113929216 z^9) EllipticK[(1/2) (1 - Sqrt[z])] + (4037880 - 2018940 Sqrt[z] + 11709852 z - 6023171 z^(3/2) + 55588148 z^2 - 28366107 z^(5/2) + 724395672 z^3 + 1065766383 z^(7/2) - 7590372372 z^4 - 4184013845 z^(9/2) + 23311381940 z^5 + 7157215680 z^(11/2) - 35007707136 z^6 - 6347937792 z^(13/2) + 28471812096 z^7 + 2875457536 z^(15/2) - 12096372736 z^8 - 528482304 z^(17/2) + 2113929216 z^9) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[3/4]^2)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["23", "4"]]], ",", FractionBox["17", "4"], ",", FractionBox["9", "2"], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["22889262825", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]]], RowBox[List["(", RowBox[List["16", " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", RowBox[List["(", RowBox[List["1009470", "+", RowBox[List["2927463", " ", "z"]], "+", RowBox[List["13897037", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["181098918", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["1897593093", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["5827845485", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["8751926784", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["7117953024", " ", SuperscriptBox["z", "7"]]], "-", RowBox[List["3024093184", " ", SuperscriptBox["z", "8"]]], "+", RowBox[List["528482304", " ", SuperscriptBox["z", "9"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], "-", RowBox[List["8", " ", RowBox[List["(", RowBox[List["1009470", "+", RowBox[List["2927463", " ", "z"]], "+", RowBox[List["13897037", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["181098918", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["1897593093", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["5827845485", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["8751926784", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["7117953024", " ", SuperscriptBox["z", "7"]]], "-", RowBox[List["3024093184", " ", SuperscriptBox["z", "8"]]], "+", RowBox[List["528482304", " ", SuperscriptBox["z", "9"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["4037880", "+", RowBox[List["2018940", " ", SqrtBox["z"]]], "+", RowBox[List["11709852", " ", "z"]], "+", RowBox[List["6023171", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["55588148", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["28366107", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["724395672", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["1065766383", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "-", RowBox[List["7590372372", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["4184013845", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]], "+", RowBox[List["23311381940", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["7157215680", " ", SuperscriptBox["z", RowBox[List["11", "/", "2"]]]]], "-", RowBox[List["35007707136", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["6347937792", " ", SuperscriptBox["z", RowBox[List["13", "/", "2"]]]]], "+", RowBox[List["28471812096", " ", SuperscriptBox["z", "7"]]], "-", RowBox[List["2875457536", " ", SuperscriptBox["z", RowBox[List["15", "/", "2"]]]]], "-", RowBox[List["12096372736", " ", SuperscriptBox["z", "8"]]], "+", RowBox[List["528482304", " ", SuperscriptBox["z", RowBox[List["17", "/", "2"]]]]], "+", RowBox[List["2113929216", " ", SuperscriptBox["z", "9"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["4037880", "-", RowBox[List["2018940", " ", SqrtBox["z"]]], "+", RowBox[List["11709852", " ", "z"]], "-", RowBox[List["6023171", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["55588148", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["28366107", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["724395672", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["1065766383", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "-", RowBox[List["7590372372", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["4184013845", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]], "+", RowBox[List["23311381940", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["7157215680", " ", SuperscriptBox["z", RowBox[List["11", "/", "2"]]]]], "-", RowBox[List["35007707136", " ", SuperscriptBox["z", "6"]]], "-", RowBox[List["6347937792", " ", SuperscriptBox["z", RowBox[List["13", "/", "2"]]]]], "+", RowBox[List["28471812096", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["2875457536", " ", SuperscriptBox["z", RowBox[List["15", "/", "2"]]]]], "-", RowBox[List["12096372736", " ", SuperscriptBox["z", "8"]]], "-", RowBox[List["528482304", " ", SuperscriptBox["z", RowBox[List["17", "/", "2"]]]]], "+", RowBox[List["2113929216", " ", SuperscriptBox["z", "9"]]]]], ")"]], " ", 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

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 23 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mn> 17 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mn> 9 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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["23", "4"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["17", "4"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[FractionBox["9", "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] </annotation> </semantics> <mo>  </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 22889262825 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 7 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 16 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 528482304 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 9 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3024093184 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 7117953024 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 8751926784 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 5827845485 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1897593093 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 181098918 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 13897037 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2927463 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1009470 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 528482304 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 9 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3024093184 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 7117953024 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 8751926784 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 5827845485 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1897593093 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 181098918 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 13897037 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2927463 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1009470 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2113929216 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 9 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 528482304 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 17 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 12096372736 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2875457536 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 15 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 28471812096 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6347937792 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 13 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 35007707136 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 7157215680 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 11 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 23311381940 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4184013845 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 9 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 7590372372 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1065766383 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 7 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 724395672 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 28366107 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 5 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 55588148 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6023171 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 11709852 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mn> 2018940 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> + </mo> <mn> 4037880 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2113929216 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 9 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 528482304 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 17 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 12096372736 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2875457536 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 15 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 28471812096 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 6347937792 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 13 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 35007707136 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 7157215680 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 11 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 23311381940 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4184013845 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 9 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 7590372372 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1065766383 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 7 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 724395672 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 28366107 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 5 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 55588148 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 6023171 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 11709852 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mn> 2018940 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> + </mo> <mn> 4037880 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 23 <sep /> 4 </cn> </apply> <cn type='rational'> 17 <sep /> 4 </cn> </list> <list> <cn type='rational'> 9 <sep /> 2 </cn> </list> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 22889262825 </cn> <apply> <power /> <pi /> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 528482304 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3024093184 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 7117953024 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8751926784 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 5827845485 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1897593093 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 181098918 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 13897037 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2927463 </cn> <ci> z </ci> </apply> <cn type='integer'> 1009470 </cn> </apply> <apply> <ci> EllipticE </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 528482304 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3024093184 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 7117953024 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8751926784 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 5827845485 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1897593093 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 181098918 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 13897037 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2927463 </cn> <ci> z </ci> </apply> <cn type='integer'> 1009470 </cn> </apply> <apply> <ci> EllipticE </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2113929216 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 528482304 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 17 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 12096372736 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2875457536 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 15 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 28471812096 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6347937792 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 13 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 35007707136 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 7157215680 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 11 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 23311381940 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4184013845 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 9 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 7590372372 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1065766383 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 724395672 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 28366107 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 55588148 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6023171 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 11709852 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2018940 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 4037880 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2113929216 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 528482304 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 17 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 12096372736 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2875457536 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 15 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 28471812096 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6347937792 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 13 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 35007707136 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 7157215680 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 11 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 23311381940 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4184013845 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 9 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 7590372372 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1065766383 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 724395672 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 28366107 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 55588148 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6023171 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 11709852 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2018940 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 4037880 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <cn type='rational'> 3 <sep /> 4 </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["23", "4"]]], ",", FractionBox["17", "4"], ",", FractionBox["9", "2"], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["16", " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", RowBox[List["(", RowBox[List["1009470", "+", RowBox[List["2927463", " ", "z"]], "+", RowBox[List["13897037", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["181098918", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["1897593093", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["5827845485", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["8751926784", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["7117953024", " ", SuperscriptBox["z", "7"]]], "-", RowBox[List["3024093184", " ", SuperscriptBox["z", "8"]]], "+", RowBox[List["528482304", " ", SuperscriptBox["z", "9"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], "-", RowBox[List["8", " ", RowBox[List["(", RowBox[List["1009470", "+", RowBox[List["2927463", " ", "z"]], "+", RowBox[List["13897037", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["181098918", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["1897593093", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["5827845485", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["8751926784", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["7117953024", " ", SuperscriptBox["z", "7"]]], "-", RowBox[List["3024093184", " ", SuperscriptBox["z", "8"]]], "+", RowBox[List["528482304", " ", SuperscriptBox["z", "9"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["4037880", "+", RowBox[List["2018940", " ", SqrtBox["z"]]], "+", RowBox[List["11709852", " ", "z"]], "+", RowBox[List["6023171", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["55588148", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["28366107", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["724395672", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["1065766383", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "-", RowBox[List["7590372372", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["4184013845", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]], "+", RowBox[List["23311381940", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["7157215680", " ", SuperscriptBox["z", RowBox[List["11", "/", "2"]]]]], "-", RowBox[List["35007707136", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["6347937792", " ", SuperscriptBox["z", RowBox[List["13", "/", "2"]]]]], "+", RowBox[List["28471812096", " ", SuperscriptBox["z", "7"]]], "-", RowBox[List["2875457536", " ", SuperscriptBox["z", RowBox[List["15", "/", "2"]]]]], "-", RowBox[List["12096372736", " ", SuperscriptBox["z", "8"]]], "+", RowBox[List["528482304", " ", SuperscriptBox["z", RowBox[List["17", "/", "2"]]]]], "+", RowBox[List["2113929216", " ", SuperscriptBox["z", "9"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["4037880", "-", RowBox[List["2018940", " ", SqrtBox["z"]]], "+", RowBox[List["11709852", " ", "z"]], "-", RowBox[List["6023171", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["55588148", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["28366107", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["724395672", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["1065766383", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "-", RowBox[List["7590372372", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["4184013845", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]], "+", RowBox[List["23311381940", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["7157215680", " ", SuperscriptBox["z", RowBox[List["11", "/", "2"]]]]], "-", RowBox[List["35007707136", " ", SuperscriptBox["z", "6"]]], "-", RowBox[List["6347937792", " ", SuperscriptBox["z", RowBox[List["13", "/", "2"]]]]], "+", RowBox[List["28471812096", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["2875457536", " ", SuperscriptBox["z", RowBox[List["15", "/", "2"]]]]], "-", RowBox[List["12096372736", " ", SuperscriptBox["z", "8"]]], "-", RowBox[List["528482304", " ", SuperscriptBox["z", RowBox[List["17", "/", "2"]]]]], "+", RowBox[List["2113929216", " ", SuperscriptBox["z", "9"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["3", "4"], "]"]], "2"]]], RowBox[List["22889262825", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["7", "/", "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]