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AppellF1






Mathematica Notation

Traditional Notation









Hypergeometric Functions > AppellF1[a,b1,b2,c,z1,z2] > Specific values > Values at fixed points > For fixed z1,z2





http://functions.wolfram.com/07.36.03.0016.01









  


  










Input Form





AppellF1[3/2, 1/2, 1, 5/2, Subscript[z, 1], Subscript[z, 2]] == (3/(Sqrt[Subscript[z, 1]] Subscript[z, 2])) ((1/Sqrt[(-Subscript[z, 1] + Subscript[z, 2])/Subscript[z, 1]]) ArcTanh[(Sqrt[Subscript[z, 1]]/Sqrt[1 - Subscript[z, 1]]) Sqrt[(-Subscript[z, 1] + Subscript[z, 2])/Subscript[z, 1]]] - ArcSin[Sqrt[Subscript[z, 1]]])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["AppellF1", "[", RowBox[List[FractionBox["3", "2"], ",", FractionBox["1", "2"], ",", "1", ",", FractionBox["5", "2"], ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]], "\[Equal]", RowBox[List[FractionBox["3", RowBox[List[SqrtBox[SubscriptBox["z", "1"]], " ", SubscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", SqrtBox[FractionBox[RowBox[List[RowBox[List["-", SubscriptBox["z", "1"]]], "+", SubscriptBox["z", "2"]]], SubscriptBox["z", "1"]]]], RowBox[List["ArcTanh", "[", RowBox[List[FractionBox[RowBox[List[SqrtBox[SubscriptBox["z", "1"]], " "]], SqrtBox[RowBox[List["1", "-", SubscriptBox["z", "1"]]]]], SqrtBox[FractionBox[RowBox[List[RowBox[List["-", SubscriptBox["z", "1"]]], "+", SubscriptBox["z", "2"]]], SubscriptBox["z", "1"]]]]], "]"]]]], "-", RowBox[List["ArcSin", "[", SqrtBox[SubscriptBox["z", "1"]], "]"]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> ; </mo> <mfrac> <mn> 5 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 3 </mn> <mrow> <msqrt> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </msqrt> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mfrac> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mfrac> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <mrow> <msqrt> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </msqrt> <mtext> </mtext> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mfrac> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> AppellF1 </ci> <cn type='rational'> 3 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='integer'> 1 </cn> <cn type='rational'> 5 <sep /> 2 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arctanh /> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arcsin /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["3", "2"], ",", FractionBox["1", "2"], ",", "1", ",", FractionBox["5", "2"], ",", SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["3", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[SqrtBox[SubscriptBox["zz", "1"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["-", SubscriptBox["zz", "1"]]], "+", SubscriptBox["zz", "2"]]], SubscriptBox["zz", "1"]]]]], SqrtBox[RowBox[List["1", "-", SubscriptBox["zz", "1"]]]]], "]"]], SqrtBox[FractionBox[RowBox[List[RowBox[List["-", SubscriptBox["zz", "1"]]], "+", SubscriptBox["zz", "2"]]], SubscriptBox["zz", "1"]]]], "-", RowBox[List["ArcSin", "[", SqrtBox[SubscriptBox["zz", "1"]], "]"]]]], ")"]]]], RowBox[List[SqrtBox[SubscriptBox["zz", "1"]], " ", SubscriptBox["zz", "2"]]]]]]]]










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





2007-05-02





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