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
LegendreP






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,mu,3,z] > General characteristics > Branch cuts > With respect to z





http://functions.wolfram.com/07.09.04.0025.01









  


  










Input Form





Limit[LegendreP[\[Nu], \[Mu], 3, x - I \[Epsilon]], \[Epsilon] -> Plus[0]] == ((2 I Pi E^(I \[Mu] Pi))/(Gamma[1 - \[Mu] + \[Nu]] Gamma[-\[Mu] - \[Nu]])) LegendreP[\[Nu], -\[Mu], 3, -x] + E^(2 I \[Mu] Pi) LegendreP[\[Nu], \[Mu], 3, x] /; x < -1










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", RowBox[List["x", "-", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]"]]]]]], "]"]], ",", RowBox[List["\[Epsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "\[Equal]", " ", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Mu]", " ", "\[Pi]"]]]]], " "]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], "]"]], " "]]], RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", RowBox[List["-", "\[Mu]"]], ",", "3", ",", RowBox[List["-", "x"]]]], "]"]]]], "+", " ", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Mu]", " ", "\[Pi]"]]], RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "x"]], "]"]]]]]]]], "/;", RowBox[List["x", "<", RowBox[List["-", "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> <mrow> <munder> <mi> lim </mi> <mrow> <mi> &#1013; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mrow> <mo> + </mo> <mn> 0 </mn> </mrow> </mrow> </munder> <mo> &#8290; </mo> <mrow> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> &#120083; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[GothicCapitalP]&quot;, LegendreQ] </annotation> </semantics> <mi> &#957; </mi> <mi> &#956; </mi> </msubsup> </mstyle> <mo> ( </mo> <semantics> <mrow> <mi> x </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#1013; </mi> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;x&quot;, &quot;-&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;\[Epsilon]&quot;]]]], HoldComplete[LegendreP, 3]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#956; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> &#120083; </mi> <annotation encoding='Mathematica'> TagBox[TagBox[&quot;\[GothicCapitalP]&quot;, LegendreQ], LegendreP] </annotation> </semantics> <mi> &#957; </mi> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <semantics> <mi> x </mi> <annotation encoding='Mathematica'> TagBox[&quot;x&quot;, HoldComplete[LegendreP, 3]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#956; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#956; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> &#120083; </mi> <annotation encoding='Mathematica'> TagBox[TagBox[&quot;\[GothicCapitalP]&quot;, LegendreQ], LegendreP] </annotation> </semantics> <mi> &#957; </mi> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> </msubsup> <mo> ( </mo> <semantics> <mrow> <mo> - </mo> <mi> x </mi> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;-&quot;, &quot;x&quot;]], HoldComplete[LegendreP, 3]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> x </mi> <mo> &lt; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <limit /> <bvar> <ci> &#1013; </ci> </bvar> <condition> <apply> <tendsto /> <ci> &#1013; </ci> <apply> <plus /> <cn type='integer'> 0 </cn> </apply> </apply> </condition> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </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> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> &#1013; </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> &#956; </ci> <pi /> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreP </ci> <apply> <ci> LegendreQ </ci> <ci> &#120083; </ci> </apply> </apply> <ci> &#957; </ci> </apply> <ci> &#956; </ci> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreP </ci> <cn type='integer'> 3 </cn> </apply> <ci> x </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> &#956; </ci> <pi /> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <ci> &#957; </ci> </apply> </apply> <apply> <ci> Gamma </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> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreP </ci> <apply> <ci> LegendreQ </ci> <ci> &#120083; </ci> </apply> </apply> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreP </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", RowBox[List["x_", "-", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]_"]]]]]], "]"]], ",", RowBox[List["\[Epsilon]_", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Mu]", " ", "\[Pi]"]]]]], ")"]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", RowBox[List["-", "\[Mu]"]], ",", "3", ",", RowBox[List["-", "x"]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], "]"]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Mu]", " ", "\[Pi]"]]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "x"]], "]"]]]]]], "/;", RowBox[List["x", "<", RowBox[List["-", "1"]]]]]]]]]]










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





2001-10-29





© 1998-2014 Wolfram Research, Inc.