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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,z] > Series representations > Generalized power series > Expansions on branch cuts > For the function itself





http://functions.wolfram.com/07.07.06.0035.01









  


  










Input Form





LegendreP[\[Nu], z] \[Proportional] (Tan[Pi \[Nu]]/2) ((Gamma[-\[Nu]]/(2^\[Nu] Gamma[1 + \[Nu]])) ((1 + x)^\[Nu] Csc[Pi \[Nu]] + 2 I (-1 - x)^\[Nu] Floor[Arg[-x + z]/(2 Pi)]) Hypergeometric2F1Regularized[-\[Nu], -\[Nu], -2 \[Nu], 2/(1 + x)] - ((2^(1 + \[Nu]) Gamma[1 + \[Nu]])/Gamma[-\[Nu]]) ((1 + x)^(-1 - \[Nu]) Csc[Pi \[Nu]] + 2 I (-1 - x)^(-1 - \[Nu]) Floor[Arg[-x + z]/(2 Pi)]) Hypergeometric2F1Regularized[1 + \[Nu], 1 + \[Nu], 2 + 2 \[Nu], 2/(1 + x)] - (1/(1 + x)) ((Gamma[1 - \[Nu]]/(2^\[Nu] Gamma[1 + \[Nu]])) ((1 + x)^\[Nu] Csc[Pi \[Nu]] + 2 I (-1 - x)^\[Nu] Floor[Arg[-x + z]/(2 Pi)]) Hypergeometric2F1Regularized[1 - \[Nu], -\[Nu], -2 \[Nu], 2/(1 + x)] - ((2^(1 + \[Nu]) Gamma[2 + \[Nu]])/ Gamma[-\[Nu]]) ((1 + x)^(-1 - \[Nu]) Csc[Pi \[Nu]] + 2 I (-1 - x)^(-1 - \[Nu]) Floor[Arg[-x + z]/(2 Pi)]) Hypergeometric2F1Regularized[1 + \[Nu], 2 + \[Nu], 2 + 2 \[Nu], 2/(1 + x)]) (z - x) + \[Ellipsis]) /; (z -> x) && Element[x, Reals] && x < -1










Standard Form





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MathML Form







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</mo> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mfrac> <mn> 2 </mn> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;\[Nu]&quot;, &quot;+&quot;, &quot;1&quot;]], Hypergeometric2F1Regularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;\[Nu]&quot;, &quot;+&quot;, &quot;1&quot;]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[RowBox[List[&quot;2&quot;, &quot; 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</mo> <mrow> <mi> x </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> x </mi> <mo> &lt; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> LegendreP </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <tan /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#957; 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</ci> </apply> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1Regularized </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <ci> x </ci> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <lt /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02





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