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variants of this functions
LegendreP




Mathematica Notation

Traditional Notation









Polynomials > LegendreP[n,mu,2,z] > Representations through more general functions > Through hypergeometric functions > Involving 2F1





http://functions.wolfram.com/05.07.26.0030.01









  


  








Input Form





LegendreP[n, \[Mu], 2, z] == (((2^(-2 - n) (Cos[2 Pi \[Mu]] - 1))/(Pi Sqrt[Pi])) (((Gamma[1 + \[Mu] + n]/(1 + 2 n)) Gamma[1/2 - n] (-z^2)^((1/2) (\[Mu] - n)) (-Csc[(1/2) Pi (\[Mu] - n)] + (Sqrt[-z^2]/z) Sec[(1/2) Pi (\[Mu] - n)]) Hypergeometric2F1[ (1/2) (1 - \[Mu] + n), (1/2) (2 - \[Mu] + n), 3/2 + n, 1/z^2])/z + 2^(2 n) Gamma[\[Mu] - n] Gamma[1/2 + n] (-z^2)^((1/2) (\[Mu] + n)) (-Csc[(1/2) Pi (\[Mu] + n)] + (Sqrt[-z^2]/z) Sec[(1/2) Pi (\[Mu] + n)]) Hypergeometric2F1[(1/2) (-\[Mu] - n), (1/2) (1 - \[Mu] - n), 1/2 - n, 1/z^2]))/(1 - z^2)^(\[Mu]/2) /; !IntervalMemberQ[Interval[{-1, 0}], z]








Standard Form





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MathML Form







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</mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ; </mo> <mrow> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </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[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot; 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</mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> sec </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> z </mi> </mfrac> <mo> - </mo> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#960; 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</mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ; </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </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[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot; &quot;, RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;-&quot;, &quot;\[Mu]&quot;]], &quot;-&quot;, &quot;n&quot;]], &quot;)&quot;]]]], Hypergeometric2F1, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot; &quot;, RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;-&quot;, &quot;\[Mu]&quot;]], &quot;-&quot;, &quot;n&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;-&quot;, &quot;\[Nu]&quot;]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[FractionBox[&quot;1&quot;, SuperscriptBox[&quot;z&quot;, &quot;2&quot;]], Hypergeometric2F1, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02