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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Hypergeometric Functions > SphericalHarmonicY[lambda,mu,theta,phi] > Integration > Definite integration > Multiple integration





http://functions.wolfram.com/07.37.21.0014.01









  


  










Input Form





Integrate[Sin[\[CurlyTheta]] SphericalHarmonicY[Subscript[n, 1], Subscript[m, 1], \[CurlyTheta], \[CurlyPhi]] SphericalHarmonicY[Subscript[n, 2], Subscript[m, 2], \[CurlyTheta], -\[CurlyPhi]], {\[CurlyTheta], 0, Pi}, {\[CurlyPhi], 0, 2 Pi}] == KroneckerDelta[Subscript[m, 1], Subscript[m, 2]] KroneckerDelta[Subscript[n, 1], Subscript[n, 2]] /; Element[Subscript[n, 1], Integers] && Subscript[n, 1] >= 0 && Element[Subscript[m, 1], Integers] && Abs[Subscript[m, 1]] <= Subscript[n, 1]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[List[SubsuperscriptBox["\[Integral]", "0", RowBox[List["2", " ", "\[Pi]"]]], RowBox[List[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["m", "1"], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n", "2"], ",", SubscriptBox["m", "2"], ",", "\[CurlyTheta]", ",", RowBox[List["-", "\[CurlyPhi]"]]]], "]"]], RowBox[List["\[DifferentialD]", "\[CurlyPhi]"]], RowBox[List["\[DifferentialD]", "\[CurlyTheta]"]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["m", "1"], ",", SubscriptBox["m", "2"]]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["n", "1"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["n", "1"], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[SubscriptBox["m", "1"], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["m", "1"], "]"]], "\[LessEqual]", SubscriptBox["n", "1"]]]]]]]]]










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> <mi> &#960; </mi> </msubsup> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msubsup> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> Y </mi> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> Y </mi> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mrow> <mo> - </mo> <mi> &#966; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> &#966; </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> &#977; </mi> </mrow> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </msup> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> </msub> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <msub> <mi> m </mi> <mn> 1 </mn> </msub> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8804; </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> &#966; </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </uplimit> <apply> <int /> <bvar> <ci> &#977; </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <pi /> </uplimit> <apply> <times /> <apply> <sin /> <ci> &#977; </ci> </apply> <apply> <ci> SphericalHarmonicY </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#977; </ci> <ci> &#966; </ci> </apply> <apply> <ci> SphericalHarmonicY </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#977; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#966; </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#8469; </ci> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <integers /> </apply> <apply> <leq /> <apply> <abs /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[List[SubsuperscriptBox["\[Integral]", "0", RowBox[List["2", " ", "\[Pi]"]]], RowBox[List[RowBox[List[RowBox[List["Sin", "[", "\[CurlyTheta]_", "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n_", "1"], ",", SubscriptBox["m_", "1"], ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n_", "2"], ",", SubscriptBox["m_", "2"], ",", "\[CurlyTheta]_", ",", RowBox[List["-", "\[CurlyPhi]_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "\[CurlyPhi]_"]], RowBox[List["\[DifferentialD]", "\[CurlyTheta]_"]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["mm", "1"], ",", SubscriptBox["mm", "2"]]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["nn", "1"], ",", SubscriptBox["nn", "2"]]], "]"]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["nn", "1"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["nn", "1"], "\[GreaterEqual]", "0"]], "&&", RowBox[List[SubscriptBox["mm", "1"], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["mm", "1"], "]"]], "\[LessEqual]", SubscriptBox["nn", "1"]]]]]]]]]]]










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





2001-10-29