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http://functions.wolfram.com/05.10.21.0009.01
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Integrate[Sin[\[CurlyTheta]] SphericalHarmonicY[n, m, \[CurlyTheta],
\[CurlyPhi]], {\[CurlyTheta], 0, Pi}, {\[CurlyPhi], 0, 2 Pi}] ==
Sqrt[4 Pi] KroneckerDelta[n, 0] KroneckerDelta[m, 0]
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Cell[BoxData[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["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], " ", RowBox[List["\[DifferentialD]", "\[CurlyPhi]"]], RowBox[List["\[DifferentialD]", "\[CurlyTheta]"]]]]]]]], "\[Equal]", RowBox[List[SqrtBox[RowBox[List["4", "\[Pi]"]]], RowBox[List["KroneckerDelta", "[", RowBox[List["n", ",", "0"]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List["m", ",", "0"]], "]"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> π </mi> </msubsup> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msubsup> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> φ </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> ϑ </mi> </mrow> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> , </mo> <mn> 0 </mn> </mrow> </msub> <mo> ⁢ </mo> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> , </mo> <mn> 0 </mn> </mrow> </msub> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> φ </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </uplimit> <apply> <int /> <bvar> <ci> ϑ </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <pi /> </uplimit> <apply> <times /> <apply> <sin /> <ci> ϑ </ci> </apply> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <ci> ϑ </ci> <ci> φ </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> KroneckerDelta </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> KroneckerDelta </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| 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["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]]]], RowBox[List["\[DifferentialD]", "\[CurlyPhi]_"]], RowBox[List["\[DifferentialD]", "\[CurlyTheta]_"]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[SqrtBox[RowBox[List["4", " ", "\[Pi]"]]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["n", ",", "0"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", ",", "0"]], "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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