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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Polynomials > SphericalHarmonicY[n,m,theta,phi] > Integration > Definite integration > Involving the direct function





http://functions.wolfram.com/05.10.21.0007.01









  


  










Input Form





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










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", RowBox[List["\[Pi]", "/", "2"]]], RowBox[List[SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], RowBox[List["m", "+", "1"]]], SuperscriptBox[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "p"], RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], RowBox[List["\[DifferentialD]", "\[CurlyTheta]"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], 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[RowBox[List[RowBox[List["p", "!"]], SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[CurlyPhi]"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["p", "+", "n", "+", "m", "+", "1"]], ")"]], "!!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["p", "-", "n", "+", "m"]], ")"]], "!!"]]]]]]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]










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> <mo> &#8747; </mo> <mn> 0 </mn> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </msubsup> <mrow> <mrow> <msup> <mi> sin </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> cos </mi> <mi> p </mi> </msup> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <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> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> &#977; </mi> </mrow> </mrow> </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> <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> <mrow> <mi> p </mi> <mo> ! </mo> </mrow> <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> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> p </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> p </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8805; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> &#977; </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </uplimit> <apply> <times /> <apply> <power /> <apply> <sin /> <ci> &#977; </ci> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <cos /> <ci> &#977; </ci> </apply> <ci> p </ci> </apply> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <ci> &#977; </ci> <ci> &#966; </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> m </ci> </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> <factorial /> <ci> p </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> m </ci> <ci> &#966; </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Factorial2 </ci> <apply> <plus /> <ci> p </ci> <ci> n </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Factorial2 </ci> <apply> <plus /> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> p </ci> <ci> &#8469; </ci> </apply> <apply> <geq /> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





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