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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Hypergeometric Functions > SphericalHarmonicY[lambda,mu,theta,phi] > Integration > Definite integration > Involving the direct function





http://functions.wolfram.com/07.37.21.0008.01









  


  










Input Form





Integrate[((Subscript[n, 1] - Subscript[n, 2]) (Subscript[n, 1] + Subscript[n, 2] + 1) - (Subscript[m, 1]^2 - Subscript[m, 2]^2)/ (1 - z^2)) SphericalHarmonicY[Subscript[n, 1], Subscript[m, 1], ArcCos[z], \[CurlyPhi]] SphericalHarmonicY[Subscript[n, 2], Subscript[m, 2], ArcCos[z], \[CurlyPhi]], {z, a, b}] == (b (Subscript[n, 1] - Subscript[n, 2]) SphericalHarmonicY[Subscript[n, 1], Subscript[m, 1], ArcCos[b], \[CurlyPhi]] SphericalHarmonicY[ Subscript[n, 2], Subscript[m, 2], ArcCos[b], \[CurlyPhi]] + (Subscript[n, 2] + Subscript[m, 2]) SphericalHarmonicY[Subscript[n, 1], Subscript[m, 1], ArcCos[b], \[CurlyPhi]] SphericalHarmonicY[ Subscript[n, 1] - 1, Subscript[m, 2], ArcCos[b], \[CurlyPhi]]) - (a (Subscript[n, 1] - Subscript[n, 2]) SphericalHarmonicY[Subscript[n, 1], Subscript[m, 1], ArcCos[a], \[CurlyPhi]] SphericalHarmonicY[ Subscript[n, 2], Subscript[m, 2], ArcCos[a], \[CurlyPhi]] + (Subscript[n, 2] + Subscript[m, 2]) SphericalHarmonicY[Subscript[n, 1], Subscript[m, 1], ArcCos[a], \[CurlyPhi]] SphericalHarmonicY[ Subscript[n, 1] - 1, Subscript[m, 2], ArcCos[a], \[CurlyPhi]]) /; Element[Subscript[n, 1], Integers] && Subscript[n, 1] >= 0 && Element[Subscript[n, 2], Integers] && Subscript[n, 2] >= 0 && Element[Subscript[m, 1], Integers] && Element[Subscript[m, 2], Integers] && Element[a, Reals] && Element[b, Reals] && Abs[Subscript[m, 1]] <= Subscript[n, 1] && Abs[Subscript[m, 2]] <= Subscript[n, 2]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "a", "b"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["n", "1"], "-", SubscriptBox["n", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[SubscriptBox["n", "1"], "+", SubscriptBox["n", "2"], "+", "1"]], ")"]]]], "-", FractionBox[RowBox[List[SubsuperscriptBox["m", "1", "2"], "-", SubsuperscriptBox["m", "2", "2"]]], RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]], ")"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["m", "1"], ",", RowBox[List["ArcCos", "[", "z", "]"]], ",", "\[CurlyPhi]"]], "]"]], RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n", "2"], ",", SubscriptBox["m", "2"], ",", RowBox[List["ArcCos", "[", "z", "]"]], ",", "\[CurlyPhi]"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[SubscriptBox["n", "1"], "-", SubscriptBox["n", "2"]]], ")"]], RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["m", "1"], ",", RowBox[List["ArcCos", "[", "b", "]"]], ",", "\[CurlyPhi]"]], "]"]], RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n", "2"], ",", SubscriptBox["m", "2"], ",", RowBox[List["ArcCos", "[", "b", "]"]], ",", "\[CurlyPhi]"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["n", "2"], "+", SubscriptBox["m", "2"]]], ")"]], RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["m", "1"], ",", RowBox[List["ArcCos", "[", "b", "]"]], ",", "\[CurlyPhi]"]], "]"]], RowBox[List["SphericalHarmonicY", "[", RowBox[List[RowBox[List[SubscriptBox["n", "1"], "-", "1"]], ",", SubscriptBox["m", "2"], ",", RowBox[List["ArcCos", "[", "b", "]"]], ",", "\[CurlyPhi]"]], "]"]]]]]], ")"]], "-", RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List[SubscriptBox["n", "1"], "-", SubscriptBox["n", "2"]]], ")"]], RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["m", "1"], ",", RowBox[List["ArcCos", "[", "a", "]"]], ",", "\[CurlyPhi]"]], "]"]], RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n", "2"], ",", SubscriptBox["m", "2"], ",", RowBox[List["ArcCos", "[", "a", "]"]], ",", "\[CurlyPhi]"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["n", "2"], "+", SubscriptBox["m", "2"]]], ")"]], RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["m", "1"], ",", RowBox[List["ArcCos", "[", "a", "]"]], ",", "\[CurlyPhi]"]], "]"]], RowBox[List["SphericalHarmonicY", "[", RowBox[List[RowBox[List[SubscriptBox["n", "1"], "-", "1"]], ",", SubscriptBox["m", "2"], ",", RowBox[List["ArcCos", "[", "a", "]"]], ",", "\[CurlyPhi]"]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["n", "1"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["n", "1"], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[SubscriptBox["n", "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["n", "2"], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[SubscriptBox["m", "1"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["m", "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List["a", "\[Element]", "Reals"]], "\[And]", RowBox[List["b", "\[Element]", "Reals"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["m", "1"], "]"]], "\[LessEqual]", SubscriptBox["n", "1"]]], "\[And]", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["m", "2"], "]"]], "\[LessEqual]", SubscriptBox["n", "2"]]]]]]]]]










MathML Form







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</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> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <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> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <msubsup> <mi> Y </mi> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </msubsup> <mo> ( </mo> <mrow> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mi> &#966; </mi> </mrow> <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> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </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> n </mi> <mn> 2 </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> <msub> <mi> m </mi> <mn> 2 </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> <mi> a </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> b </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </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> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8804; </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <lowlimit> <ci> a </ci> </lowlimit> <uplimit> <ci> b </ci> </uplimit> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <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> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </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> <apply> <arccos /> <ci> z </ci> </apply> <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> <apply> <arccos /> <ci> z </ci> </apply> <ci> &#966; </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <plus /> <apply> <times /> <ci> b </ci> <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> <apply> <arccos /> <ci> b </ci> </apply> <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> <apply> <arccos /> <ci> b </ci> </apply> <ci> &#966; </ci> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> SphericalHarmonicY </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <arccos /> <ci> b </ci> </apply> <ci> &#966; </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> <apply> <arccos /> <ci> b </ci> </apply> <ci> &#966; </ci> </apply> <apply> <plus /> <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> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <times /> <ci> a </ci> <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> <apply> <arccos /> <ci> a </ci> </apply> <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> <apply> <arccos /> <ci> a </ci> </apply> <ci> &#966; </ci> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> SphericalHarmonicY </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <arccos /> <ci> a </ci> </apply> <ci> &#966; </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> <apply> <arccos /> <ci> a </ci> </apply> <ci> &#966; </ci> </apply> <apply> <plus /> <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> </apply> </apply> </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> n </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8469; </ci> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <integers /> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <integers /> </apply> <apply> <in /> <ci> a </ci> <reals /> </apply> <apply> <in /> <ci> b </ci> <reals /> </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> <leq /> <apply> <abs /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </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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