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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Hypergeometric Functions > SphericalHarmonicY[lambda,mu,theta,phi] > Integral representations > On the real axis > Of the direct function





http://functions.wolfram.com/07.37.07.0002.01









  


  










Input Form





SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] == (-1)^m E^(I m \[CurlyPhi]) Sqrt[((2 n + 1)/(4 Pi)) ((n + m)!/(n - m)!)] (Csc[\[CurlyTheta]]^m/(m - 1)!) Integrate[LegendreP[n, t] (t - Cos[\[CurlyTheta]])^(m - 1), {t, Cos[\[CurlyTheta]], 1}] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m >= 0 && m <= n










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[CurlyPhi]"]]], " ", SqrtBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["2", "n"]], "+", "1"]], RowBox[List["4", "\[Pi]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], "!"]]]]]], " ", FractionBox[SuperscriptBox[RowBox[List["Csc", "[", "\[CurlyTheta]", "]"]], "m"], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]], " ", RowBox[List[UnderoverscriptBox["\[Integral]", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "1"], RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n", ",", "t"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["t", "-", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], ")"]], RowBox[List["m", "-", "1"]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[LessEqual]", "n"]]]]]]]]










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> <msubsup> <mi> Y </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <mi> &#966; </mi> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mfrac> <mrow> <msup> <mi> csc </mi> <mi> m </mi> </msup> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> <mn> 1 </mn> </msubsup> <mrow> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> t </mi> <mo> - </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8804; </mo> <mi> n </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <ci> &#977; </ci> <ci> &#966; </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> m </ci> <ci> &#966; </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <power /> <apply> <csc /> <ci> &#977; </ci> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <apply> <cos /> <ci> &#977; </ci> </apply> </lowlimit> <uplimit> <cn type='integer'> 1 </cn> </uplimit> <apply> <times /> <apply> <ci> LegendreP </ci> <ci> n </ci> <ci> t </ci> </apply> <apply> <power /> <apply> <plus /> <ci> t </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <ci> &#977; </ci> </apply> </apply> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <in /> <ci> m </ci> <ci> &#8469; </ci> </apply> <apply> <leq /> <ci> m </ci> <ci> n </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[CurlyPhi]"]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], "!"]]]], RowBox[List[RowBox[List["(", RowBox[List["4", " ", "\[Pi]"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], "!"]]]]]], " ", SuperscriptBox[RowBox[List["Csc", "[", "\[CurlyTheta]", "]"]], "m"], " ", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "1"], RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n", ",", "t"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["t", "-", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], ")"]], RowBox[List["m", "-", "1"]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[LessEqual]", "n"]]]]]]]]]]










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





2001-10-29





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