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variants of this functions
LegendreP




Mathematica Notation

Traditional Notation









Polynomials > LegendreP[n,z] > Complex characteristics > Imaginary part





http://functions.wolfram.com/05.03.19.0002.01









  


  








Input Form





Im[LegendreP[n, x + I y]] == Sum[(((-1)^j 2^(2 j + 1) Pochhammer[1/2, 2 j + 1])/(2 j + 1)!) GegenbauerC[n - 2 j - 1, 2 j + 3/2, x] y^(2 j + 1), {j, 0, Floor[(n - 1)/2]}] /; Element[x, Reals] && Element[y, Reals]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Im", "[", RowBox[List["LegendreP", "[", RowBox[List["n", ",", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]]]], "]"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "/", "2"]], "]"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], SuperscriptBox["2", RowBox[List[RowBox[List["2", "j"]], "+", "1"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "/", "2"]], ",", RowBox[List[RowBox[List["2", "j"]], "+", "1"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "j"]], "+", "1"]], ")"]], "!"]]], " ", RowBox[List["GegenbauerC", "[", RowBox[List[RowBox[List["n", "-", RowBox[List["2", "j"]], "-", "1"]], ",", RowBox[List[RowBox[List["2", "j"]], "+", RowBox[List["3", "/", "2"]]]], ",", "x"]], "]"]], " ", SuperscriptBox["y", RowBox[List[RowBox[List["2", "j"]], "+", "1"]]]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["y", "\[Element]", "Reals"]]]]]]]]








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> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> </msub> <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> <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> <mtext> </mtext> <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> <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> </mfrac> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <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;, FractionBox[&quot;1&quot;, &quot;2&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> <mrow> <msubsup> <mi> C </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> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <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> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <imaginary /> <apply> <ci> LegendreP </ci> <ci> n </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> <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> <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> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> GegenbauerC </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 /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='rational'> 3 <sep /> 2 </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> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Im", "[", RowBox[List["LegendreP", "[", RowBox[List["n_", ",", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]]], "]"]]]], ")"]], " ", RowBox[List["GegenbauerC", "[", RowBox[List[RowBox[List["n", "-", RowBox[List["2", " ", "j"]], "-", "1"]], ",", RowBox[List[RowBox[List["2", " ", "j"]], "+", FractionBox["3", "2"]]], ",", "x"]], "]"]], " ", SuperscriptBox["y", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["y", "\[Element]", "Reals"]]]]]]]]]]


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





2001-10-29