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variants of this functions
SphericalHarmonicY




Mathematica Notation

Traditional Notation









Hypergeometric Functions > SphericalHarmonicY[lambda,mu,theta,phi] > Series representations > Generalized power series > Expansions at theta==Pi





http://functions.wolfram.com/07.37.06.0025.01









  


  








Input Form





SphericalHarmonicY[\[Lambda], 0, \[CurlyTheta], \[CurlyPhi]] \[Proportional] (Sin[Pi \[Lambda]]/2) Sqrt[(2 \[Lambda] + 1)/Pi^3] (Log[(\[CurlyTheta] - Pi)^2/4] (1 + (\[CurlyTheta] - Pi)^2) + 2 EulerGamma + PolyGamma[-\[Lambda]] + PolyGamma[\[Lambda] + 1] + O[(\[CurlyTheta] - Pi)^2]) /; (\[CurlyTheta] -> Pi)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "0", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Lambda]"]], "]"]], "2"], SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], SuperscriptBox["\[Pi]", "3"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Log", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"], "4"], "]"]], RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"]]], ")"]]]], "+", RowBox[List["2", " ", "EulerGamma"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["-", "\[Lambda]"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["\[Lambda]", "+", "1"]], "]"]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"], "]"]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List["\[CurlyTheta]", "\[Rule]", "\[Pi]"]], ")"]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mi> Y </mi> <mi> &#955; </mi> <mn> 0 </mn> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mfrac> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <msup> <mi> &#960; </mi> <mn> 3 </mn> </msup> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#955; </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#977; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> SphericalHarmonicY </ci> <ci> &#955; </ci> <cn type='integer'> 0 </cn> <ci> &#977; </ci> <ci> &#966; </ci> </apply> <apply> <times /> <apply> <times /> <apply> <sin /> <apply> <times /> <pi /> <ci> &#955; </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <power /> <pi /> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <ln /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> &#977; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> &#977; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#955; </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> &#955; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <eulergamma /> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> &#977; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> &#977; </ci> <pi /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]_", ",", "0", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Lambda]"]], "]"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], SuperscriptBox["\[Pi]", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], ")"]], "2"]]], ")"]]]], "+", RowBox[List["2", " ", "EulerGamma"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["-", "\[Lambda]"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["\[Lambda]", "+", "1"]], "]"]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["\[CurlyTheta]", "-", "\[Pi]"]], "]"]], "2"]]], ")"]]]], "/;", RowBox[List["(", RowBox[List["\[CurlyTheta]", "\[Rule]", "\[Pi]"]], ")"]]]]]]]]


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





2001-10-29