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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Polynomials > SphericalHarmonicY[n,m,theta,phi] > Representations through more general functions > Through hypergeometric functions > Involving 2F1





http://functions.wolfram.com/05.10.26.0009.01









  


  










Input Form





SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] == (-1)^((m/2) (1 + Sign[m])) E^(I m \[CurlyPhi]) Sqrt[((2 n + 1)/Pi) ((n + Abs[m])!/(n - Abs[m])!)] (1/(2^(Abs[m] + 1) Abs[m]!)) Cos[\[CurlyTheta]/2]^(2 n - 2 Abs[m]) (Sin[\[CurlyTheta]]^2)^(Abs[m]/2) Hypergeometric2F1[-n + Abs[m], -n, Abs[m] + 1, -Tan[\[CurlyTheta]/2]^2]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[FractionBox["m", "2"], RowBox[List["(", RowBox[List["1", "+", RowBox[List["Sign", "[", "m", "]"]]]], ")"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[CurlyPhi]"]]], " ", SqrtBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["2", "n"]], "+", "1"]], "\[Pi]"], " ", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", "+", RowBox[List["Abs", "[", "m", "]"]]]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["Abs", "[", "m", "]"]]]], ")"]], "!"]]]]]], FractionBox["1", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "+", "1"]]], RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "!"]]]]], " ", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], RowBox[List[RowBox[List["2", "n"]], "-", RowBox[List["2", RowBox[List["Abs", "[", "m", "]"]]]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], "2"], ")"]], RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "/", "2"]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "+", RowBox[List["Abs", "[", "m", "]"]]]], ",", RowBox[List["-", "n"]], ",", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "+", "1"]], ",", RowBox[List["-", SuperscriptBox[RowBox[List["Tan", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"]]]]], "]"]]]]]]]]










MathML Form







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</annotation> </semantics> <mi> m </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> m </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> m </mi> <semantics> <mo> &#10072; 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</mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mrow> <msup> <mi> tan </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> &#977; </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List[&quot;\[LeftBracketingBar]&quot;, &quot;m&quot;, &quot;\[RightBracketingBar]&quot;]], &quot;-&quot;, &quot;n&quot;]], Hypergeometric2F1, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, &quot;n&quot;]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[RowBox[List[&quot;\[LeftBracketingBar]&quot;, &quot;m&quot;, &quot;\[RightBracketingBar]&quot;]], &quot;+&quot;, &quot;1&quot;]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, RowBox[List[SuperscriptBox[&quot;tan&quot;, &quot;2&quot;], &quot;(&quot;, FractionBox[&quot;\[CurlyTheta]&quot;, &quot;2&quot;], &quot;)&quot;]]]], Hypergeometric2F1, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <ci> &#977; 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</ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <abs /> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <sin /> <ci> &#977; </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <abs /> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <apply> <abs /> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <plus /> <apply> <abs /> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <tan /> <apply> <times /> <ci> &#977; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </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]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[FractionBox["1", "2"], " ", "m", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Sign", "[", "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", "+", RowBox[List["Abs", "[", "m", "]"]]]], ")"]], "!"]]]], RowBox[List["\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["Abs", "[", "m", "]"]]]], ")"]], "!"]]]]]], " ", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", RowBox[List["Abs", "[", "m", "]"]]]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], "2"], ")"]], FractionBox[RowBox[List["Abs", "[", "m", "]"]], "2"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "+", RowBox[List["Abs", "[", "m", "]"]]]], ",", RowBox[List["-", "n"]], ",", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "+", "1"]], ",", RowBox[List["-", SuperscriptBox[RowBox[List["Tan", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"]]]]], "]"]]]], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "+", "1"]]], " ", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "!"]]]]]]]]]










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





2001-10-29





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