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
LegendreP




Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,mu,3,z] > Representations through more general functions > Through Meijer G > Classical cases involving products of Legendre P





http://functions.wolfram.com/07.09.26.0067.01









  


  








Input Form





LegendreP[\[Nu], \[Mu], 3, Sqrt[1 + z^2]] LegendreP[\[Mu] - 1/2, \[Nu] + 1/2, 3, Sqrt[1 + z^2]/z] == (1/(Sqrt[2] Pi Gamma[-\[Mu] - \[Nu]])) MeijerG[{{3/4, 5/4 + \[Nu]}, {1/4 - \[Nu]}}, {{1/4, 1/4 - \[Mu]}, {\[Mu] + 1/4}}, z, 1/2] /; Re[z] > 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]], "]"]], RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Mu]", "-", FractionBox["1", "2"]]], ",", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ",", "3", ",", FractionBox[SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]], "z"]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[SqrtBox["2"], "\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], "]"]]]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", RowBox[List[FractionBox["5", "4"], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "4"], "-", "\[Nu]"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", RowBox[List[FractionBox["1", "4"], "-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["\[Mu]", "+", FractionBox["1", "4"]]], "}"]]]], "}"]], ",", "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> <mrow> <msubsup> <semantics> <mi> &#120083; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[GothicCapitalP]&quot;, LegendreP] </annotation> </semantics> <mi> &#957; </mi> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <semantics> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <annotation encoding='Mathematica'> TagBox[SqrtBox[RowBox[List[SuperscriptBox[&quot;z&quot;, &quot;2&quot;], &quot;+&quot;, &quot;1&quot;]]], HoldComplete[LegendreP, 3]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> &#120083; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[GothicCapitalP]&quot;, LegendreP] </annotation> </semantics> <mrow> <mi> &#956; </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mrow> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msubsup> <mo> ( </mo> <semantics> <mfrac> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mi> z </mi> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox[SqrtBox[RowBox[List[SuperscriptBox[&quot;z&quot;, &quot;2&quot;], &quot;+&quot;, &quot;1&quot;]]], &quot;z&quot;], HoldComplete[LegendreP, 3]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 3 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 2 </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> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 5 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mi> &#957; </mi> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> , </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </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;3&quot;, &quot;,&quot;, &quot;3&quot;]], RowBox[List[&quot;2&quot;, &quot;,&quot;, &quot;2&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[&quot;3&quot;, &quot;4&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;\[Nu]&quot;, &quot;+&quot;, FractionBox[&quot;5&quot;, &quot;4&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;4&quot;], &quot;-&quot;, &quot;\[Nu]&quot;]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[FractionBox[&quot;1&quot;, &quot;4&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;4&quot;], &quot;-&quot;, &quot;\[Mu]&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;\[Mu]&quot;, &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;4&quot;]]], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreP </ci> <ci> &#120083; </ci> </apply> <ci> &#957; </ci> </apply> <ci> &#956; </ci> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreP </ci> <cn type='integer'> 3 </cn> </apply> <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> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreP </ci> <ci> &#120083; </ci> </apply> <apply> <plus /> <ci> &#956; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreP </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <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> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <pi /> <apply> <times /> <ci> &#915; </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list> <cn type='rational'> 3 <sep /> 4 </cn> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 5 <sep /> 4 </cn> </apply> </list> <list> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </list> </list> <list> <list> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> </list> <list> <apply> <plus /> <ci> &#956; </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </list> </list> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <gt /> <apply> <times /> <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[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z_", "2"]]]]]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Mu]_", "-", FractionBox["1", "2"]]], ",", RowBox[List["\[Nu]_", "+", FractionBox["1", "2"]]], ",", "3", ",", FractionBox[SqrtBox[RowBox[List["1", "+", SuperscriptBox["z_", "2"]]]], "z_"]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", RowBox[List[FractionBox["5", "4"], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "4"], "-", "\[Nu]"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", RowBox[List[FractionBox["1", "4"], "-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["\[Mu]", "+", FractionBox["1", "4"]]], "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]], RowBox[List[SqrtBox["2"], " ", "\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], "]"]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]]


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





2001-10-29