|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/07.37.06.0030.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]] ==
E^(I \[CurlyPhi] \[Mu]) ((Sin[Pi (\[Lambda] - \[Mu])] Sin[Pi \[Lambda]]
Sqrt[2 \[Lambda] + 1] Sqrt[Gamma[\[Lambda] - \[Mu] + 1]]
Sqrt[Gamma[\[Lambda] + \[Mu] + 1]])/(2^(\[Lambda] + 1) Pi^2
Gamma[3/2 + \[Lambda]])) (Cos[\[CurlyTheta]] - 1)^(-\[Lambda] - 1)
((Cos[\[CurlyTheta]/2]^2)^(\[Mu]/2)/(Sin[\[CurlyTheta]/2]^2)^(\[Mu]/2))
Sum[((Pochhammer[\[Lambda] + 1, k] Pochhammer[\[Lambda] + \[Mu] + 1, k])/
(k! Pochhammer[2 \[Lambda] + 2, k])) (PolyGamma[k + 1] -
PolyGamma[-k - \[Lambda] - \[Mu]] - PolyGamma[\[Lambda] + k + 1] +
PolyGamma[2 \[Lambda] + k + 2]) (2/(1 - Cos[\[CurlyTheta]]))^k,
{k, 0, Infinity}] + ((2^(\[Lambda] + 1) Sin[Pi (\[Mu] - \[Lambda])]
Gamma[\[Lambda] + \[Mu] + 1])/(Pi Gamma[-\[Lambda]]
Gamma[2 \[Lambda] + 2])) (Cos[\[CurlyTheta]] - 1)^(-\[Lambda] - 1)
((Cos[\[CurlyTheta]/2]^2)^(\[Mu]/2)/(Sin[\[CurlyTheta]/2]^2)^(\[Mu]/2))
Log[(Cos[\[CurlyTheta]] - 1)/2] Hypergeometric2F1[\[Lambda] + 1,
\[Lambda] + \[Mu] + 1, 2 \[Lambda] + 2, 2/(1 - Cos[\[CurlyTheta]])] +
(1/(2^\[Lambda] Gamma[\[Lambda] + 1])) (Cos[\[CurlyTheta]] - 1)^\[Lambda]
((Cos[\[CurlyTheta]/2]^2)^(\[Mu]/2)/(Sin[\[CurlyTheta]/2]^2)^(\[Mu]/2))
Sum[(((2 \[Lambda] - k)! Pochhammer[-\[Lambda], k])/
(k! Gamma[1 - k - \[Mu] + \[Lambda]])) (2/(1 - Cos[\[CurlyTheta]]))^k,
{k, 0, 2 \[Lambda]}] /; Element[2 \[Lambda] + 1, Integers] &&
2 \[Lambda] + 1 >= 0 && !Element[\[Lambda] - \[Mu], Integers]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]", ",", "\[Mu]", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", " ", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "\[Mu]"]]], " ", FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "-", "\[Mu]"]], ")"]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Lambda]"]], "]"]], SqrtBox[RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "1"]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]]]], RowBox[List[SuperscriptBox["2", RowBox[List["\[Lambda]", "+", "1"]]], SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Lambda]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "1"]]], " ", 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["\[Mu]", "/", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Lambda]", "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "2"]], ",", "k"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "k"]], "-", "\[Lambda]", "-", "\[Mu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["\[Lambda]", "+", "k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "k", "+", "2"]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]]], ")"]], "k"]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Lambda]", "+", "1"]]], RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Lambda]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Lambda]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "2"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "1"]]], " ", 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["\[Mu]", "/", "2"]]]], RowBox[List["Log", "[", FractionBox[RowBox[List[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "-", "1"]], "2"], "]"]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["\[Lambda]", "+", "1"]], ",", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], ",", RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "2"]], ",", FractionBox["2", RowBox[List["1", "-", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]]]]], "]"]]]], " ", "+", RowBox[List[FractionBox[SuperscriptBox["2", RowBox[List["-", "\[Lambda]"]]], RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "1"]], "]"]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "-", "1"]], ")"]], "\[Lambda]"], " ", 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["\[Mu]", "/", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["2", " ", "\[Lambda]"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "-", "k"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Lambda]"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "k", "-", "\[Mu]", "+", "\[Lambda]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]]], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "1"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List["\[Lambda]", "-", "\[Mu]"]], "\[Element]", "Integers"]], "]"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<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> λ </mi> <mi> μ </mi> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mtext> </mtext> <mrow> <mrow> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> φ </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> - </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> - </mo> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> </mrow> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> λ </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mfrac> <msup> <mrow> <msup> <mi> cos </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> ϑ </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> μ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> ϑ </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> μ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Lambda]", "+", "1"]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], ")"]], "k"], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "2"]], ")"]], "k"], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> λ </mi> <mo> - </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mi> λ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> λ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> λ </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mfrac> <msup> <mrow> <msup> <mi> cos </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> ϑ </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> μ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> ϑ </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> μ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mfrac> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Lambda]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "2"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["2", RowBox[List["1", "-", RowBox[List["cos", "(", "\[CurlyTheta]", ")"]]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> λ </mi> </mrow> </msup> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> λ </mi> </msup> <mo> ⁢ </mo> <mfrac> <msup> <mrow> <msup> <mi> cos </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> ϑ </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> μ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> ϑ </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> μ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> </munderover> <mrow> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> λ </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Lambda]"]], ")"]], "k"], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> λ </mi> <mo> - </mo> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> λ </mi> <mo> - </mo> <mi> μ </mi> </mrow> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> SphericalHarmonicY </ci> <ci> λ </ci> <ci> μ </ci> <ci> ϑ </ci> <ci> φ </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> φ </ci> <ci> μ </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <apply> <plus /> <ci> λ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <ci> λ </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> λ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> λ </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> λ </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <cos /> <ci> ϑ </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <power /> <apply> <cos /> <apply> <times /> <ci> ϑ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> μ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> ϑ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> μ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> λ </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> 2 </cn> </apply> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <ci> ϑ </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> λ </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <cos /> <ci> ϑ </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <power /> <apply> <cos /> <apply> <times /> <ci> ϑ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> μ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> ϑ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> μ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <times /> <apply> <plus /> <apply> <cos /> <ci> ϑ </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> λ </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <ci> ϑ </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <cos /> <ci> ϑ </ci> </apply> <cn type='integer'> -1 </cn> </apply> <ci> λ </ci> </apply> <apply> <times /> <apply> <power /> <apply> <power /> <apply> <cos /> <apply> <times /> <ci> ϑ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> μ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> ϑ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> μ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> λ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <ci> ϑ </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> ℕ </ci> </apply> <apply> <notin /> <apply> <plus /> <ci> λ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["\[Lambda]_", ",", "\[Mu]_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "\[Mu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "-", "\[Mu]"]], ")"]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Lambda]"]], "]"]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Lambda]", "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "k"]], "-", "\[Lambda]", "-", "\[Mu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["\[Lambda]", "+", "k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "k", "+", "2"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "2"]], ",", "k"]], "]"]]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Lambda]", "+", "1"]]], " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Lambda]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Lambda]", "+", "1"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Lambda]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "-", "1"]], ")"]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["\[Lambda]", "+", "1"]], ",", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], ",", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "2"]], ",", FractionBox["2", RowBox[List["1", "-", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Lambda]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "2"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Lambda]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]], "-", "1"]], ")"]], "\[Lambda]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["2", " ", "\[Lambda]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "-", "k"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Lambda]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "k", "-", "\[Mu]", "+", "\[Lambda]"]], "]"]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "1"]], "]"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sin", "[", FractionBox["\[CurlyTheta]", "2"], "]"]], "2"], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List["!", RowBox[List[RowBox[List["\[Lambda]", "-", "\[Mu]"]], "\[Element]", "Integers"]]]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
){kind=small}
