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variants of this functions
JacobiP




Mathematica Notation

Traditional Notation









Hypergeometric Functions > JacobiP[nu,a,b,z] > Complex characteristics > Imaginary part





http://functions.wolfram.com/07.15.19.0002.01









  


  








Input Form





Im[JacobiP[n, a, b, x + I y]] == Sum[(((-1)^j Pochhammer[n + a + b + 1, 2 j + 1])/(2^(2 j + 1) (2 j + 1)!)) JacobiP[n - 2 j - 1, a + 2 j + 1, b + 2 j + 1, x] y^(2 j + 1), {j, 0, Floor[(n - 1)/2]}] /; Element[x, Reals] && Element[y, Reals] && Element[a, Reals] && Element[b, Reals] && Element[n, Integers] && n >= 0








Standard Form





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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> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msubsup> <mi> P </mi> <mi> n </mi> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;b&quot;, &quot;+&quot;, &quot;n&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;j&quot;]], &quot;+&quot;, &quot;1&quot;]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mi> y </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mi> P </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> y </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> a </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> b </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <imaginary /> <apply> <ci> JacobiP </ci> <ci> n </ci> <ci> a </ci> <ci> b </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <ci> y </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> JacobiP </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> j </ci> </apply> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> x </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <in /> <ci> y </ci> <reals /> </apply> <apply> <in /> <ci> a </ci> <reals /> </apply> <apply> <in /> <ci> b </ci> <reals /> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29