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variants of this functions
SphericalHarmonicY




Mathematica Notation

Traditional Notation









Hypergeometric Functions > SphericalHarmonicY[lambda,mu,theta,phi] > Representations through more general functions > Through hypergeometric functions > Involving 2F1





http://functions.wolfram.com/07.37.26.0010.01









  


  








Input Form





SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] == (-1)^(n - m) (-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]!)) Sin[\[CurlyTheta]/2]^(2 n - 2 Abs[m]) (Sin[\[CurlyTheta]]^2)^(Abs[m]/2) Hypergeometric2F1[-n + Abs[m], -n, Abs[m] + 1, -Cot[\[CurlyTheta]/2]^2] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && Abs[m] <= n








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "m"]]], 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["Sin", "[", 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["Cot", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "\[LessEqual]", "n"]]]]]]]]








MathML Form







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</mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <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> <mi> m </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8804; </mo> <mi> n </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <ci> &#977; </ci> <ci> &#966; </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <ci> m </ci> <apply> <plus /> <apply> <ci> Sign </ci> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> m </ci> <ci> &#966; </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <abs /> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <apply> <abs /> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <abs /> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> &#977; </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> <cot /> <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> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> <apply> <leq /> <apply> <abs /> <ci> m </ci> </apply> <ci> n </ci> </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]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "m"]]], " ", 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["Sin", "[", 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["Cot", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"]]]]], "]"]]]], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "+", "1"]]], " ", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "!"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "\[LessEqual]", "n"]]]]]]]]]]


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





2001-10-29