|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/07.23.03.0385.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Hypergeometric2F1[4/5, 1, 9/5, z] == (1/(5 z^(4/5)))
(Log[1 - z] - 5 Log[1 - z^(1/5)] +
Sqrt[5] Log[1 + ((Sqrt[5] + 1)/2) z^(1/5) + z^(2/5)] -
Sqrt[5] Log[1 - ((Sqrt[5] - 1)/2) z^(1/5) + z^(2/5)] -
2 Sqrt[10 - 2 Sqrt[5]] ArcTan[4 + (Sqrt[5] + 1) z^(1/5),
Sqrt[10 - 2 Sqrt[5]] z^(1/5)] - 2 Sqrt[10 + 2 Sqrt[5]]
ArcTan[4 - (Sqrt[5] - 1) z^(1/5), Sqrt[10 + 2 Sqrt[5]] z^(1/5)])
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["4", "5"], ",", "1", ",", FractionBox["9", "5"], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["5", " ", SuperscriptBox["z", RowBox[List["4", "/", "5"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]], "-", RowBox[List["5", RowBox[List["Log", "[", RowBox[List["1", "-", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]], "]"]]]], "+", RowBox[List[SqrtBox["5"], RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List[FractionBox[RowBox[List[SqrtBox["5"], "+", "1"]], "2"], SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]], "+", SuperscriptBox["z", RowBox[List["2", "/", "5"]]]]], "]"]]]], "-", RowBox[List[SqrtBox["5"], RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List[FractionBox[RowBox[List[SqrtBox["5"], "-", "1"]], "2"], SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]], "+", SuperscriptBox["z", RowBox[List["2", "/", "5"]]]]], "]"]]]], "-", RowBox[List["2", SqrtBox[RowBox[List["10", "-", RowBox[List["2", SqrtBox["5"]]]]]], RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["4", "+", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["5"], "+", "1"]], ")"]], SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]]]], ",", RowBox[List[SqrtBox[RowBox[List["10", "-", RowBox[List["2", SqrtBox["5"]]]]]], SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]]]], "]"]]]], "-", RowBox[List["2", SqrtBox[RowBox[List["10", "+", RowBox[List["2", SqrtBox["5"]]]]]], RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["4", "-", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["5"], "-", "1"]], ")"]], SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]]]], ",", RowBox[List[SqrtBox[RowBox[List["10", "+", RowBox[List["2", SqrtBox["5"]]]]]], SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]]]], "]"]]]]]], ")"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<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> <mfrac> <mn> 4 </mn> <mn> 5 </mn> </mfrac> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mfrac> <mn> 9 </mn> <mn> 5 </mn> </mfrac> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["4", "5"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["9", "5"], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 4 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msqrt> <mn> 5 </mn> </msqrt> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mfrac> <mrow> <msqrt> <mn> 5 </mn> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mtext> </mtext> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> </mrow> <mo> + </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msqrt> <mn> 5 </mn> </msqrt> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <mrow> <msqrt> <mn> 5 </mn> </msqrt> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> </mrow> <mo> + </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 10 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msqrt> <mn> 5 </mn> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> </mrow> </mrow> <mo> , </mo> <mrow> <msqrt> <mrow> <mn> 10 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 10 </mn> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msqrt> <mn> 5 </mn> </msqrt> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> </mrow> </mrow> <mo> , </mo> <mrow> <msqrt> <mrow> <mn> 10 </mn> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Hypergeometric2F1 </ci> <cn type='rational'> 4 <sep /> 5 </cn> <cn type='integer'> 1 </cn> <cn type='rational'> 9 <sep /> 5 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 4 <sep /> 5 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 5 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 5 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 2 <sep /> 5 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 5 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 2 <sep /> 5 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 10 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arctan /> <apply> <plus /> <cn type='integer'> 4 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 5 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 10 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 5 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 10 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arctan /> <apply> <plus /> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 5 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 10 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 5 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["4", "5"], ",", "1", ",", FractionBox["9", "5"], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]], "-", RowBox[List["5", " ", RowBox[List["Log", "[", RowBox[List["1", "-", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]], "]"]]]], "+", RowBox[List[SqrtBox["5"], " ", RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SqrtBox["5"], "+", "1"]], ")"]], " ", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]], "+", SuperscriptBox["z", RowBox[List["2", "/", "5"]]]]], "]"]]]], "-", RowBox[List[SqrtBox["5"], " ", RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SqrtBox["5"], "-", "1"]], ")"]], " ", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]], "+", SuperscriptBox["z", RowBox[List["2", "/", "5"]]]]], "]"]]]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["10", "-", RowBox[List["2", " ", SqrtBox["5"]]]]]], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["4", "+", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["5"], "+", "1"]], ")"]], " ", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]]]], ",", RowBox[List[SqrtBox[RowBox[List["10", "-", RowBox[List["2", " ", SqrtBox["5"]]]]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["10", "+", RowBox[List["2", " ", SqrtBox["5"]]]]]], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["4", "-", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["5"], "-", "1"]], ")"]], " ", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]]]], ",", RowBox[List[SqrtBox[RowBox[List["10", "+", RowBox[List["2", " ", SqrtBox["5"]]]]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]]]], "]"]]]]]], RowBox[List["5", " ", SuperscriptBox["z", RowBox[List["4", "/", "5"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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,...,ap},{b1,...,bq},z] | |
|
|
|