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 SphericalHarmonicY

 http://functions.wolfram.com/05.10.06.0019.01

 Input Form

 SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] \[Proportional] (Sqrt[(2 n + 1)/(4 Pi)] (Sqrt[Gamma[n - m + 1]]/Sqrt[Gamma[n + m + 1]]) 2^m E^(I \[CurlyPhi] m) (1/Gamma[1 - m] + (1/12) (-((3 n (n + 1))/Gamma[2 - m]) + 1/Gamma[-m]) \[CurlyTheta]^2 + (1/1440) ((30 n (n + 1) (m + 1))/Gamma[2 - m] + (45 (n - 1) n (n + 1) (n + 2))/Gamma[3 - m] + (7 - 5 m)/Gamma[-m]) \[CurlyTheta]^4 + (1/362880) (-((63 n (n + 1) (4 + m (3 + 5 m)))/ Gamma[2 - m]) - (945 (n - 1) n (n + 1) (n + 2) (m + 2))/ Gamma[3 - m] - (945 (n - 1) n (n + 1) (n + 2) (n^2 + n - 6))/ Gamma[4 - m] + (124 + 7 m (5 m - 21))/Gamma[-m]) \[CurlyTheta]^6 + O[\[CurlyTheta]^8]))/(\[CurlyTheta]^2)^(m/2) /; (\[CurlyTheta] -> 0)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Proportional]", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", "n"]], "+", "1"]], RowBox[List["4", "\[Pi]"]]]], FractionBox[SqrtBox[RowBox[List["Gamma", "[", RowBox[List["n", "-", "m", "+", "1"]], "]"]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["n", "+", "m", "+", "1"]], "]"]]]], SuperscriptBox["2", "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "m"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[CurlyTheta]", "2"], ")"]], RowBox[List[RowBox[List["-", "m"]], "/", "2"]]], RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["1", "-", "m"]], "]"]]], "+", RowBox[List[FractionBox["1", "12"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["3", " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "m"]], "]"]]]]], "+", FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["-", "m"]], "]"]]]]], ")"]], SuperscriptBox["\[CurlyTheta]", "2"]]], " ", "+", RowBox[List[FractionBox["1", "1440"], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["30", " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "m"]], "]"]]], "+", FractionBox[RowBox[List["45", " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "2"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["3", "-", "m"]], "]"]]], "+", FractionBox[RowBox[List["7", "-", RowBox[List["5", " ", "m"]]]], RowBox[List["Gamma", "[", RowBox[List["-", "m"]], "]"]]]]], ")"]], SuperscriptBox["\[CurlyTheta]", "4"]]], " ", "+", RowBox[List[FractionBox["1", "362880"], RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["63", " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["5", "m"]]]], ")"]]]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "m"]], "]"]]]]], "-", FractionBox[RowBox[List["945", " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["m", "+", "2"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["3", "-", "m"]], "]"]]], "-", FractionBox[RowBox[List["945", " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "2"]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["n", "2"], "+", "n", "-", "6"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["4", "-", "m"]], "]"]]], "+", FractionBox[RowBox[List["124", "+", RowBox[List["7", " ", "m", " ", RowBox[List["(", RowBox[List[RowBox[List["5", " ", "m"]], "-", "21"]], ")"]]]]]], RowBox[List["Gamma", "[", RowBox[List["-", "m"]], "]"]]]]], ")"]], SuperscriptBox["\[CurlyTheta]", "6"]]], " ", "+", RowBox[List["O", "[", SuperscriptBox["\[CurlyTheta]", "8"], "]"]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List["\[CurlyTheta]", "\[Rule]", "0"]], ")"]]]]]]

 MathML Form

 Y n m ( ϑ , φ ) 2 n + 1 4 π Γ ( n - m + 1 ) Γ ( n + m + 1 ) 2 m φ m ( ϑ 2 ) - m / 2 ( 1 Γ ( 1 - m ) + 1 12 ( - 3 n ( n + 1 ) Γ ( 2 - m ) + 1 Γ ( - m ) ) ϑ 2 + 1 1440 ( 30 n ( n + 1 ) ( m + 1 ) Γ ( 2 - m ) + 45 ( n - 1 ) n ( n + 1 ) ( n + 2 ) Γ ( 3 - m ) + 7 - 5 m Γ ( - m ) ) ϑ 4 + 1 362880 ( - 63 n ( n + 1 ) ( 4 + m ( 3 + 5 m ) ) Γ ( 2 - m ) - 945 ( n - 1 ) n ( n + 1 ) ( n + 2 ) ( m + 2 ) Γ ( 3 - m ) - 945 ( n - 1 ) n ( n + 1 ) ( n + 2 ) ( n 2 + n - 6 ) Γ ( 4 - m ) + 124 + 7 m ( 5 m - 21 ) Γ ( - m ) ) ϑ 6 + O ( ϑ 8 ) ) /; ( ϑ "\[Rule]" 0 ) Condition Proportional SphericalHarmonicY n m ϑ φ 2 n 1 4 -1 1 2 Gamma n -1 m 1 1 2 Gamma n m 1 1 2 -1 2 m φ m ϑ 2 -1 m 2 -1 1 Gamma 1 -1 m -1 1 12 -1 3 n n 1 Gamma 2 -1 m -1 1 Gamma -1 m -1 ϑ 2 1 1440 30 n n 1 m 1 Gamma 2 -1 m -1 45 n -1 n n 1 n 2 Gamma 3 -1 m -1 7 -1 5 m Gamma -1 m -1 ϑ 4 1 362880 -1 63 n n 1 4 m 3 5 m Gamma 2 -1 m -1 -1 945 n -1 n n 1 n 2 m 2 Gamma 3 -1 m -1 -1 945 n -1 n n 1 n 2 n 2 n -6 Gamma 4 -1 m -1 124 7 m 5 m -21 Gamma -1 m -1 ϑ 6 O ϑ 8 Rule ϑ 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], RowBox[List["4", " ", "\[Pi]"]]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["n", "-", "m", "+", "1"]], "]"]]], " ", SuperscriptBox["2", "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]", " ", "m"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[CurlyTheta]", "2"], ")"]], RowBox[List["-", FractionBox["m", "2"]]]], " ", RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["1", "-", "m"]], "]"]]], "+", RowBox[List[FractionBox["1", "12"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["3", " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "m"]], "]"]]]]], "+", FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["-", "m"]], "]"]]]]], ")"]], " ", SuperscriptBox["\[CurlyTheta]", "2"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["30", " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "m"]], "]"]]], "+", FractionBox[RowBox[List["45", " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "2"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["3", "-", "m"]], "]"]]], "+", FractionBox[RowBox[List["7", "-", RowBox[List["5", " ", "m"]]]], RowBox[List["Gamma", "[", RowBox[List["-", "m"]], "]"]]]]], ")"]], " ", SuperscriptBox["\[CurlyTheta]", "4"]]], "1440"], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["63", " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["5", " ", "m"]]]], ")"]]]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "m"]], "]"]]]]], "-", FractionBox[RowBox[List["945", " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["m", "+", "2"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["3", "-", "m"]], "]"]]], "-", FractionBox[RowBox[List["945", " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "2"]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["n", "2"], "+", "n", "-", "6"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["4", "-", "m"]], "]"]]], "+", FractionBox[RowBox[List["124", "+", RowBox[List["7", " ", "m", " ", RowBox[List["(", RowBox[List[RowBox[List["5", " ", "m"]], "-", "21"]], ")"]]]]]], RowBox[List["Gamma", "[", RowBox[List["-", "m"]], "]"]]]]], ")"]], " ", SuperscriptBox["\[CurlyTheta]", "6"]]], "362880"], "+", SuperscriptBox[RowBox[List["O", "[", "\[CurlyTheta]", "]"]], "8"]]], ")"]]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["n", "+", "m", "+", "1"]], "]"]]]], "/;", RowBox[List["(", RowBox[List["\[CurlyTheta]", "\[Rule]", "0"]], ")"]]]]]]]]

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

 2001-10-29

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