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

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 8 and fixed z and a<0

For fixed z and a=-31/8, b>=a

For fixed z and a=-31/8, b=-19/8

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

Input Form

Hypergeometric2F1[-(31/8), -(19/8), 4, z] == (2048 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (1734016 - 31469681 z + 401600815 z^2 + 9756872030 z^3 + 14030330390 z^4 + 3154043699 z^5 + 4610331 z^6) EllipticE[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (1734016 - 31469681 z + 401600815 z^2 + 9756872030 z^3 + 14030330390 z^4 + 3154043699 z^5 + 4610331 z^6) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[1 - z] (1734016 - 31469681 z + 401600815 z^2 + 9756872030 z^3 + 14030330390 z^4 + 3154043699 z^5 + 4610331 z^6) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - (1734016 - 32119937 z + 413224141 z^2 - 8455985710 z^3 - 29821175590 z^4 - 15777560941 z^5 - 982000503 z^6 + 18441324 z^7) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (18498959203725 Pi (1 + Sqrt[1 - z])^(1/4) z^3)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["31", "8"]]], ",", RowBox[List["-", FractionBox["19", "8"]]], ",", "4", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["2048", " ", SuperscriptBox["2", RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List["1734016", "-", RowBox[List["31469681", " ", "z"]], "+", RowBox[List["401600815", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["9756872030", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["14030330390", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["3154043699", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["4610331", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], "-", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List["1734016", "-", RowBox[List["31469681", " ", "z"]], "+", RowBox[List["401600815", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["9756872030", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["14030330390", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["3154043699", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["4610331", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], "-", RowBox[List[SqrtBox[RowBox[List["1", "-", "z"]]], " ", RowBox[List["(", RowBox[List["1734016", "-", RowBox[List["31469681", " ", "z"]], "+", RowBox[List["401600815", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["9756872030", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["14030330390", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["3154043699", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["4610331", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1734016", "-", RowBox[List["32119937", " ", "z"]], "+", RowBox[List["413224141", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["8455985710", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["29821175590", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["15777560941", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["982000503", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["18441324", " ", SuperscriptBox["z", "7"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["18498959203725", " ", "\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox["z", "3"]]], ")"]]]]]]]]

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> 31 </mn> <mn> 8 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mn> 19 </mn> <mn> 8 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mn> 4 </mn> <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["31", "8"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", FractionBox["19", "8"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox["4", 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> <mrow> <mo> ( </mo> <mrow> <mn> 2048 </mn> <mo> ⁢ </mo> <mroot> <mn> 2 </mn> <mn> 4 </mn> </mroot> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 4 </mn> </mroot> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4610331 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3154043699 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 14030330390 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 9756872030 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 401600815 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 31469681 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1734016 </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> <mfrac> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 4 </mn> </mroot> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4610331 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3154043699 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 14030330390 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 9756872030 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 401600815 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 31469681 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1734016 </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> <mfrac> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 4 </mn> </mroot> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 4 </mn> </mroot> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4610331 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3154043699 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 14030330390 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 9756872030 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 401600815 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 31469681 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1734016 </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> <mfrac> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 4 </mn> </mroot> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 18441324 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 982000503 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 15777560941 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 29821175590 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 8455985710 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 413224141 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 32119937 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1734016 </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> <mfrac> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 4 </mn> </mroot> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mn> 18498959203725 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mroot> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mroot> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </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'> 31 <sep /> 8 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 19 <sep /> 8 </cn> </apply> </list> <list> <cn type='integer'> 4 </cn> </list> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2048 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4610331 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3154043699 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 14030330390 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 9756872030 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 401600815 </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'> 31469681 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 1734016 </cn> </apply> <apply> <ci> EllipticE </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4610331 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3154043699 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 14030330390 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 9756872030 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 401600815 </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'> 31469681 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 1734016 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4610331 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3154043699 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 14030330390 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 9756872030 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 401600815 </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'> 31469681 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 1734016 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 18441324 </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'> 982000503 </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'> 15777560941 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 29821175590 </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'> 8455985710 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 413224141 </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'> 32119937 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 1734016 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 18498959203725 </cn> <pi /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["31", "8"]]], ",", RowBox[List["-", FractionBox["19", "8"]]], ",", "4", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["2048", " ", SuperscriptBox["2", RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List["1734016", "-", RowBox[List["31469681", " ", "z"]], "+", RowBox[List["401600815", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["9756872030", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["14030330390", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["3154043699", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["4610331", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], "-", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List["1734016", "-", RowBox[List["31469681", " ", "z"]], "+", RowBox[List["401600815", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["9756872030", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["14030330390", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["3154043699", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["4610331", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], "-", RowBox[List[SqrtBox[RowBox[List["1", "-", "z"]]], " ", RowBox[List["(", RowBox[List["1734016", "-", RowBox[List["31469681", " ", "z"]], "+", RowBox[List["401600815", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["9756872030", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["14030330390", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["3154043699", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["4610331", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1734016", "-", RowBox[List["32119937", " ", "z"]], "+", RowBox[List["413224141", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["8455985710", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["29821175590", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["15777560941", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["982000503", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["18441324", " ", SuperscriptBox["z", "7"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List["18498959203725", " ", "\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]], ")"]], RowBox[List["1", "/", "4"]]], " ", 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]