Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
Hypergeometric2F1Regularized




Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric2F1Regularized[a,b,c,z] > Representations through more general functions > Through Meijer G > Generalized cases involving algebraic functions and 2F1with algebraic arguments





http://functions.wolfram.com/07.24.26.0261.01









  


  








Input Form





(z + Sqrt[1 + z^2])^(1 - c) Hypergeometric2F1Regularized[a, b, c, (z - Sqrt[1 + z^2])/(2 z)] Hypergeometric2F1[1 + a - c, 1 + b - c, 1 + a + b - c, (z - Sqrt[1 + z^2])/(2 z)] == ((2^(a + b - c) Gamma[1 + a + b - c])/(Sqrt[Pi] Gamma[a] Gamma[b])) MeijerG[{{(3 - c)/2}, {a + b + (1 - c)/2, a + b + (3 - 3 c)/2, (1 + c)/2}}, {{a + (1 - c)/2, b + (1 - c)/2, (1/2) (1 + a + b - c), 1 + (1/2) (a + b - c)}, {}}, z, 1/2] /; Re[z] > 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]], ")"]], RowBox[List["1", "-", "c"]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", "b", ",", "c", ",", FractionBox[RowBox[List["z", "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]], RowBox[List["2", " ", "z"]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", "a", "-", "c"]], ",", RowBox[List["1", "+", "b", "-", "c"]], ",", RowBox[List["1", "+", "a", "+", "b", "-", "c"]], ",", FractionBox[RowBox[List["z", "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]], RowBox[List["2", " ", "z"]]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["a", "+", "b", "-", "c"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "+", "b", "-", "c"]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", "b", "]"]]]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["3", "-", "c"]], "2"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["a", "+", "b", "+", FractionBox[RowBox[List["1", "-", "c"]], "2"]]], ",", RowBox[List["a", "+", "b", "+", FractionBox[RowBox[List["3", "-", RowBox[List["3", "c"]]]], "2"]]], ",", FractionBox[RowBox[List["1", "+", "c"]], "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["a", "+", FractionBox[RowBox[List["1", "-", "c"]], "2"]]], ",", RowBox[List["b", "+", FractionBox[RowBox[List["1", "-", "c"]], "2"]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "-", "c"]], ")"]]]], ",", RowBox[List["1", "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "+", "b", "-", "c"]], ")"]]]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> c </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ; </mo> <mi> c </mi> <mo> ; </mo> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;a&quot;, Hypergeometric2F1Regularized, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;b&quot;, Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[&quot;c&quot;, Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[FractionBox[RowBox[List[&quot;z&quot;, &quot;-&quot;, SqrtBox[RowBox[List[SuperscriptBox[&quot;z&quot;, &quot;2&quot;], &quot;+&quot;, &quot;1&quot;]]]]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;z&quot;]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;a&quot;, &quot;-&quot;, &quot;c&quot;, &quot;+&quot;, &quot;1&quot;]], Hypergeometric2F1, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;b&quot;, &quot;-&quot;, &quot;c&quot;, &quot;+&quot;, &quot;1&quot;]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;b&quot;, &quot;-&quot;, &quot;c&quot;, &quot;+&quot;, &quot;1&quot;]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[FractionBox[RowBox[List[&quot;z&quot;, &quot;-&quot;, SqrtBox[RowBox[List[SuperscriptBox[&quot;z&quot;, &quot;2&quot;], &quot;+&quot;, &quot;1&quot;]]]]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;z&quot;]]], Hypergeometric2F1, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 4 </mn> <mo> , </mo> <mn> 4 </mn> </mrow> <mrow> <mn> 4 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn> 3 </mn> <mo> - </mo> <mi> c </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> c </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> c </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> c </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;4&quot;, &quot;,&quot;, &quot;4&quot;]], RowBox[List[&quot;4&quot;, &quot;,&quot;, &quot;1&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[RowBox[List[TagBox[&quot;z&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], MeijerG, Rule[Editable, True]]]], MeijerG], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[FractionBox[RowBox[List[&quot;3&quot;, &quot;-&quot;, &quot;c&quot;]], &quot;2&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;b&quot;, &quot;+&quot;, FractionBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;c&quot;]], &quot;2&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;b&quot;, &quot;+&quot;, FractionBox[RowBox[List[&quot;3&quot;, &quot;-&quot;, RowBox[List[&quot;3&quot;, &quot;c&quot;]]]], &quot;2&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;c&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;2&quot;], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[RowBox[List[&quot;a&quot;, &quot;+&quot;, FractionBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;c&quot;]], &quot;2&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;b&quot;, &quot;+&quot;, FractionBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;c&quot;]], &quot;2&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;b&quot;, &quot;-&quot;, &quot;c&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;2&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;b&quot;, &quot;-&quot;, &quot;c&quot;]], &quot;2&quot;], &quot;+&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1Regularized </ci> <ci> a </ci> <ci> b </ci> <ci> c </ci> <apply> <times /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list> <apply> <times /> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> c </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> </list> <list> <list> <apply> <plus /> <ci> a </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <list /> </list> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z_", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z_", "2"]]]]]], ")"]], RowBox[List["1", "-", "c_"]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a_", ",", "b_", ",", "c_", ",", FractionBox[RowBox[List["z_", "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z_", "2"]]]]]], RowBox[List["2", " ", "z_"]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", "a_", "-", "c_"]], ",", RowBox[List["1", "+", "b_", "-", "c_"]], ",", RowBox[List["1", "+", "a_", "+", "b_", "-", "c_"]], ",", FractionBox[RowBox[List["z_", "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z_", "2"]]]]]], RowBox[List["2", " ", "z_"]]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["a", "+", "b", "-", "c"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "+", "b", "-", "c"]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["3", "-", "c"]], "2"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["a", "+", "b", "+", FractionBox[RowBox[List["1", "-", "c"]], "2"]]], ",", RowBox[List["a", "+", "b", "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["3", " ", "c"]]]], ")"]]]]]], ",", FractionBox[RowBox[List["1", "+", "c"]], "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["a", "+", FractionBox[RowBox[List["1", "-", "c"]], "2"]]], ",", RowBox[List["b", "+", FractionBox[RowBox[List["1", "-", "c"]], "2"]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "-", "c"]], ")"]]]], ",", RowBox[List["1", "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "+", "b", "-", "c"]], ")"]]]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", "b", "]"]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]]


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





2001-10-29