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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,mu,2,z] > Series representations > Generalized power series > Expansions at z==-1





http://functions.wolfram.com/07.08.06.0016.01









  


  










Input Form





LegendreP[\[Nu], \[Mu], 2, z] == (Gamma[-\[Mu]]/(2^(\[Mu]/2) (Gamma[-\[Mu] - \[Nu]] Gamma[1 - \[Mu] + \[Nu]]))) (z + 1)^(\[Mu]/2) (1 + ((\[Mu] (1 + \[Mu]) - 2 \[Nu] (1 + \[Nu]))/(4 (1 + \[Mu]))) (z + 1) + ((\[Mu] (1 + \[Mu]) (2 + \[Mu])^2 - 4 (2 + \[Mu] (2 + \[Mu])) \[Nu] - 4 (1 + \[Mu])^2 \[Nu]^2 + 8 \[Nu]^3 + 4 \[Nu]^4)/ (32 (1 + \[Mu]) (2 + \[Mu]))) (z + 1)^2 + \[Ellipsis]) - ((2^(\[Mu]/2) Sin[Pi \[Nu]] Gamma[\[Mu]])/((z + 1)^(\[Mu]/2) Pi)) (1 + ((\[Mu] (1 - \[Mu]) + 2 \[Nu] (1 + \[Nu]))/(4 (-1 + \[Mu]))) (z + 1) + ((-5 \[Mu]^3 + \[Mu]^4 + 4 (-1 + \[Nu]) \[Nu] (1 + \[Nu]) (2 + \[Nu]) - 4 \[Mu]^2 (-2 + \[Nu] + \[Nu]^2) + \[Mu] (-4 + 8 \[Nu] (1 + \[Nu])))/(32 (-2 + \[Mu]) (-1 + \[Mu]))) (z + 1)^2 + \[Ellipsis]) /; Abs[(z + 1)/2] < 1 && !Element[\[Mu], Integers]










Standard Form





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MathML Form







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</mi> <mn> 4 </mn> </msup> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> &#956; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 32 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> + </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; 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</ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <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> <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> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> &#956; </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> &#956; </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> &#956; 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</ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> &#956; </ci> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> <apply> <ci> Gamma </ci> <ci> &#956; </ci> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> &#956; </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -5 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <ci> &#957; </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> &#957; </ci> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <cn type='integer'> -2 </cn> <ci> &#957; </ci> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <plus /> <cn type='integer'> -4 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <ci> &#957; </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <plus /> <cn type='integer'> -2 </cn> <ci> &#956; </ci> </apply> <apply> <plus /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <abs /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <notin /> <ci> &#956; </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





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