Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

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
Hypergeometric2F1






Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric2F1[a,b,c,z] > Representations through more general functions > Through Meijer G > Generalized cases involving algebraic functions with squares in arguments





http://functions.wolfram.com/07.23.26.0217.02









  


  










Input Form





(z + Sqrt[1 + z^2])^(a - 2 c + 2) Hypergeometric2F1[a, 1, c, (z - Sqrt[1 + z^2])/(2 z)] == ((c - 1)/Sqrt[Pi]) MeijerG[{{2 + a/2 - b}, {1 + a/2, -(a/2) + c}}, {{1, 1/2, a/2}, {}}, 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["a", "-", RowBox[List["2", " ", "c"]], "+", "2"]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List["a", ",", "1", ",", "c", ",", FractionBox[RowBox[List["z", "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]], RowBox[List["2", "z"]]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["c", "-", "1"]], SqrtBox["\[Pi]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", "+", FractionBox["a", "2"], "-", "b"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox["a", "2"]]], ",", RowBox[List[RowBox[List["-", FractionBox["a", "2"]]], "+", "c"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox["1", "2"], ",", FractionBox["a", "2"]]], "}"]], ",", 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> <mi> a </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> </msup> <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> <mi> a </mi> <mo> , </mo> <mn> 1 </mn> </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[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;a&quot;, Hypergeometric2F1, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;1&quot;, Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[&quot;c&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> <mi> c </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <msqrt> <mi> &#960; </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 3 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> <mrow> <mn> 3 </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> <mrow> <mfrac> <mi> a </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> a </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> c </mi> <mo> - </mo> <mfrac> <mi> a </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 1 </mn> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mi> a </mi> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;3&quot;, &quot;,&quot;, &quot;3&quot;]], RowBox[List[&quot;3&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[RowBox[List[FractionBox[&quot;a&quot;, &quot;2&quot;], &quot;-&quot;, &quot;b&quot;, &quot;+&quot;, &quot;2&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;a&quot;, &quot;2&quot;], &quot;+&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;c&quot;, &quot;-&quot;, FractionBox[&quot;a&quot;, &quot;2&quot;]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[&quot;1&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;a&quot;, &quot;2&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 /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <ci> a </ci> <cn type='integer'> 1 </cn> <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> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <cn type='integer'> 2 </cn> </apply> </list> <list> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </list> </list> <list> <list> <cn type='integer'> 1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <ci> a </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </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["a_", "-", RowBox[List["2", " ", "c_"]], "+", "2"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["a_", ",", "1", ",", "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["c", "-", "1"]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", "+", FractionBox["a", "2"], "-", "b"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox["a", "2"]]], ",", RowBox[List[RowBox[List["-", FractionBox["a", "2"]]], "+", "c"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox["1", "2"], ",", FractionBox["a", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]], SqrtBox["\[Pi]"]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]]










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





2001-10-29





© 1998-2014 Wolfram Research, Inc.