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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Polynomials > SphericalHarmonicY[n,m,theta,phi] > Summation > Infinite summation





http://functions.wolfram.com/05.10.23.0011.01









  


  










Input Form





Sum[(((n + m - p)! (n - m + p - 1)!)/(n! Sqrt[(2 n + 1) (n - m)! (n + m)!])) SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] w^(n - m), {n, m, Infinity}] == (((2 m - p)! (p - 1)!)/(2^(m + 1) Sqrt[Pi] m!^2)) ((-Sin[\[CurlyTheta]]) E^(I \[CurlyPhi]))^m Hypergeometric2F1[p, 2 m - p + 1, m + 1, (1 - w - Sqrt[1 - 2 w Cos[\[CurlyTheta]] + w^2])/2] Hypergeometric2F1[p, 2 m - p + 1, m + 1, (1 + w - Sqrt[1 - 2 w Cos[\[CurlyTheta]] + w^2])/2] /; m >= 0 && Element[p, Integers] && p >= 0 && p <= 2 m && Element[\[CurlyTheta], Reals] && Element[\[CurlyPhi], Reals] && Abs[w] < 1










Standard Form





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MathML Form







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</ci> </apply> </apply> </apply> <apply> <power /> <ci> w </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <geq /> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <ci> p </ci> <ci> &#8469; </ci> </apply> <apply> <leq /> <ci> p </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> </apply> <apply> <in /> <ci> &#977; </ci> <reals /> </apply> <apply> <in /> <ci> &#966; </ci> <reals /> </apply> <apply> <lt /> <apply> <abs /> <ci> w </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n_", "=", "m_"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n_", "+", "m_", "-", "p_"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n_", "-", "m_", "+", "p_", "-", "1"]], ")"]], "!"]]]], ")"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], " ", SuperscriptBox["w_", RowBox[List["n_", "-", "m_"]]]]], RowBox[List[RowBox[List["n_", "!"]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n_"]], "+", "1"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n_", "-", "m_"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n_", "+", "m_"]], ")"]], "!"]]]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m"]], "-", "p"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["p", "-", "1"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]"]]]]], ")"]], "m"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["p", ",", RowBox[List[RowBox[List["2", " ", "m"]], "-", "p", "+", "1"]], ",", RowBox[List["m", "+", "1"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "w", "-", SqrtBox[RowBox[List["1", "-", RowBox[List["2", " ", "w", " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "+", SuperscriptBox["w", "2"]]]]]], ")"]]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["p", ",", RowBox[List[RowBox[List["2", " ", "m"]], "-", "p", "+", "1"]], ",", RowBox[List["m", "+", "1"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "w", "-", SqrtBox[RowBox[List["1", "-", RowBox[List["2", " ", "w", " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "+", SuperscriptBox["w", "2"]]]]]], ")"]]]]]], "]"]]]], RowBox[List[SuperscriptBox["2", RowBox[List["m", "+", "1"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "!"]], ")"]], "2"]]]], "/;", RowBox[List[RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "0"]], "&&", RowBox[List["p", "\[LessEqual]", RowBox[List["2", " ", "m"]]]], "&&", RowBox[List["\[CurlyTheta]", "\[Element]", "Reals"]], "&&", RowBox[List["\[CurlyPhi]", "\[Element]", "Reals"]], "&&", RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "<", "1"]]]]]]]]]]










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





2001-10-29





© 1998- Wolfram Research, Inc.