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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 Meijer G > Classical cases involving unit step theta





http://functions.wolfram.com/05.07.26.0016.01









  


  








Input Form





(UnitStep[1 - Abs[z]]/Sqrt[1 - z]) (LegendreP[n, \[Mu], 2, -Sqrt[1 - z]] + LegendreP[n, \[Mu], 2, Sqrt[1 - z]]) == ((2^(\[Mu] + 1) Pi)/(Gamma[(1/2) (1 - \[Mu] - n)] Gamma[(1/2) (2 - \[Mu] + n)])) MeijerG[{{}, {-(n/2), (1 + n)/2}}, {{-(\[Mu]/2), \[Mu]/2}, {}}, z] /; !IntervalMemberQ[Interval[{-1, 0}], z]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[FractionBox[RowBox[List["UnitStep", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", "z", "]"]]]], "]"]], SqrtBox[RowBox[List["1", "-", "z"]]]], RowBox[List["(", RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n", ",", "\[Mu]", ",", "2", ",", RowBox[List["-", SqrtBox[RowBox[List["1", "-", "z"]]]]]]], "]"]], "+", RowBox[List["LegendreP", "[", RowBox[List["n", ",", "\[Mu]", ",", "2", ",", SqrtBox[RowBox[List["1", "-", "z"]]]]], "]"]]]], ")"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Mu]", "+", "1"]]], "\[Pi]"]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]", "-", "n"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "\[Mu]", "+", "n"]], ")"]]]], "]"]], " "]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["n", "2"]]], ",", FractionBox[RowBox[List["1", "+", "n"]], "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]








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> <mfrac> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <semantics> <mrow> <mo> - </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;-&quot;, SqrtBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;z&quot;]]]]], HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> <mi> &#956; 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</mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;2&quot;, &quot;,&quot;, &quot;2&quot;]], RowBox[List[&quot;2&quot;, &quot;,&quot;, &quot;0&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[&quot;z&quot;, MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;n&quot;, &quot;2&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;n&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;2&quot;], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </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 encoding='MathML-Content'> <mrow> <mrow> <mrow> <mfrac> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <semantics> <mrow> <mo> - </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;-&quot;, SqrtBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;z&quot;]]]]], HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <semantics> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <annotation encoding='Mathematica'> TagBox[SqrtBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;z&quot;]]], HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mrow> <mtext> </mtext> <mrow> <mi> &#915; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#956; </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#915; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> &#956; </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;2&quot;, &quot;,&quot;, &quot;2&quot;]], RowBox[List[&quot;2&quot;, &quot;,&quot;, &quot;0&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[&quot;z&quot;, MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;n&quot;, &quot;2&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;n&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;2&quot;], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </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





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", "z_", "]"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "\[Mu]_", ",", "2", ",", RowBox[List["-", SqrtBox[RowBox[List["1", "-", "z_"]]]]]]], "]"]], "+", RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "\[Mu]_", ",", "2", ",", SqrtBox[RowBox[List["1", "-", "z_"]]]]], "]"]]]], ")"]]]], SqrtBox[RowBox[List["1", "-", "z_"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Mu]", "+", "1"]]], " ", "\[Pi]"]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["n", "2"]]], ",", FractionBox[RowBox[List["1", "+", "n"]], "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]", "-", "n"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "\[Mu]", "+", "n"]], ")"]]]], "]"]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]


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





2001-10-29