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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 generic point z==z0 > For the function itself





http://functions.wolfram.com/07.08.06.0044.01









  


  










Input Form





LegendreP[\[Nu], \[Mu], 2, z] == ((Sin[Pi \[Nu]]/(Pi Gamma[-\[Mu] - \[Nu]] Gamma[1 - \[Mu] + \[Nu]])) (1/(Subscript[z, 0] + 1))^((1/2) \[Mu] Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) (Subscript[z, 0] + 1)^ ((1/2) \[Mu] (Floor[Arg[z - Subscript[z, 0]]/(2 Pi)] + 1)) Sum[((((-1)^(k - j) 2^(i - k))/((i - j)! j! (k - i)!)) Pochhammer[-(\[Mu]/2), j] Pochhammer[\[Mu]/2, i - j] (1 - Subscript[z, 0])^(j - i) ((2 Pi I (-1)^k Floor[Arg[z - Subscript[z, 0]]/(2 Pi)] Floor[(Arg[Subscript[z, 0] + 1] + Pi)/(2 Pi)] Gamma[-i + k - \[Nu]] Gamma[i + \[Nu] + 1] Hypergeometric2F1Regularized[-i + k - \[Nu], -i + k + \[Nu] + 1, -i + k + \[Mu] + 1, (Subscript[z, 0] + 1)/2])/ E^(I Pi \[Mu] Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) - ((-1)^i MeijerG[{{i - k + \[Nu] + 1, i - k - \[Nu]}, {}}, {{0, i - k - \[Mu]}, {}}, (Subscript[z, 0] + 1)/2])/ ((1/(Subscript[z, 0] + 1))^(\[Mu] Floor[Arg[z - Subscript[z, 0]]/ (2 Pi)]) (Subscript[z, 0] + 1)^(\[Mu] Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]))) (z - Subscript[z, 0])^k)/ (Subscript[z, 0] + 1)^j, {k, 0, Infinity}, {i, 0, k}, {j, 0, i}])/ ((1/(1 - Subscript[z, 0]))^((1/2) \[Mu] Floor[Arg[Subscript[z, 0] - z]/(2 Pi)]) (1 - Subscript[z, 0])^ ((1/2) \[Mu] (Floor[Arg[Subscript[z, 0] - z]/(2 Pi)] + 1)))










Standard Form





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MathML Form







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</mo> <mi> &#956; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> i </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mi> j </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mi> i </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> i </mi> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> i </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;]]], &quot;)&quot;]], &quot;j&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> i </mi> <mo> - </mo> <mi> j </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;], &quot;)&quot;]], RowBox[List[&quot;i&quot;, &quot;-&quot;, &quot;j&quot;]]], Pochhammer] </annotation> </semantics> <mo> &#8290; 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Rule Form





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





2007-05-02





© 1998- Wolfram Research, Inc.