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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Hypergeometric Functions > SphericalHarmonicY[lambda,mu,theta,phi] > Differentiation > Low-order differentiation > With respect to mu





http://functions.wolfram.com/07.37.20.0007.01









  


  










Input Form





D[SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]], \[Mu]] == (-(1/2)) (Log[Sin[\[CurlyTheta]/2]^2] - Log[Cos[\[CurlyTheta]/2]^2] - 2 I \[CurlyPhi] - 2 PolyGamma[1 - \[Mu]] + PolyGamma[1 + \[Lambda] - \[Mu]] + PolyGamma[1 + \[Lambda] + \[Mu]]) SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]] + ((\[Lambda] (\[Lambda] + 1))/((\[Mu] - 1) Gamma[2 - \[Mu]])) Sqrt[(2 \[Lambda] + 1)/(4 Pi)] (Sqrt[Gamma[\[Lambda] - \[Mu] + 1]]/ Sqrt[Gamma[\[Lambda] + \[Mu] + 1]]) E^(I \[CurlyPhi] \[Mu]) ((Cos[\[CurlyTheta]/2]^2)^(\[Mu]/2)/(Sin[\[CurlyTheta]/2]^2)^ (\[Mu]/2 - 2)) HypergeometricPFQ[{{1 - \[Lambda], 2 + \[Lambda]}, {1}, {1, 1 - \[Mu]}}, {{2, 2 - \[Mu]}, {}, {2 - \[Mu]}}, Sin[\[CurlyTheta]/2]^2, Sin[\[CurlyTheta]/2]^2]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "\[Mu]"], RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], "]"]], "-", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], "]"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[CurlyPhi]"]], "-", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "-", "\[Mu]"]], "]"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Lambda]", "-", "\[Mu]"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Lambda]", "+", "\[Mu]"]], "]"]]]], ")"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["\[Lambda]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "1"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["\[Mu]", "-", "1"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "-", "\[Mu]"]], "]"]]]]], SqrtBox[FractionBox[RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "1"]], RowBox[List["4", "\[Pi]"]]]], FractionBox[SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "\[Mu]"]]], " ", FractionBox[SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List[RowBox[List["\[Mu]", "/", "2"]], "-", "2"]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Lambda]"]], ",", RowBox[List["2", "+", "\[Lambda]"]]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["2", "-", "\[Mu]"]], "}"]]]], "}"]], ",", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"]]], "]"]]]]]]]]]]










MathML Form







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</ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> Y </ms> <ms> &#955; </ms> <ms> &#956; </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#977; </ms> <ms> , </ms> <ms> &#966; </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["\[Mu]_"]]], RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]_", ",", "\[Mu]_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], "]"]], "-", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], "]"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[CurlyPhi]"]], "-", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "-", "\[Mu]"]], "]"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Lambda]", "-", "\[Mu]"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Lambda]", "+", "\[Mu]"]], "]"]]]], ")"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Lambda]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "1"]], ")"]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], RowBox[List["4", " ", "\[Pi]"]]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "\[Mu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Lambda]"]], ",", RowBox[List["2", "+", "\[Lambda]"]]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["2", "-", "\[Mu]"]], "}"]]]], "}"]], ",", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["\[Mu]", "-", "1"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "-", "\[Mu]"]], "]"]]]], ")"]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List[FractionBox["\[Mu]", "2"], "-", "2"]]]]]]]]]]]]










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





2001-10-29





© 1998- Wolfram Research, Inc.