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variants of this functions
LegendreP




Mathematica Notation

Traditional Notation









Polynomials > LegendreP[n,z] > Integral representations > On the real axis > Of the direct function





http://functions.wolfram.com/05.03.07.0001.01









  


  








Input Form





LegendreP[n, z] == (1/Pi) Integrate[(z - Sqrt[z^2 - 1] Cos[t])^n, {t, 0, Pi}]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["\[Pi]", " "]]], RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]], " ", RowBox[List["Cos", "[", "t", "]"]]]]]], ")"]], "n"], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> &#960; </mi> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mi> &#960; </mi> </msubsup> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> LegendreP </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <pi /> </uplimit> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <cos /> <ci> t </ci> </apply> </apply> </apply> </apply> <ci> n </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]], " ", RowBox[List["Cos", "[", "t", "]"]]]]]], ")"]], "n"], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Pi]"]]]]]


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





2001-10-29