|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/05.10.06.0027.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] ==
(Sqrt[((2 n + 1) (n + m)! (n - m)!)/(4 Pi)]
Sum[(1/(i! j! k! 2^i 2^j)) (KroneckerDelta[i + j + k, n]
KroneckerDelta[i - j, m] (x - I y)^j (-x - I y)^i z^k), {i, 0, n},
{j, 0, n}, {k, 0, n}])/Sqrt[x^2 + y^2 + z^2]^n /;
Element[n, Integers] && n >= 0 && Element[m, Integers] && -n <= m <= n &&
x == Cos[\[CurlyPhi]] Sin[\[CurlyTheta]] &&
y == Sin[\[CurlyPhi]] Sin[\[CurlyTheta]] && z == Cos[\[CurlyTheta]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", SuperscriptBox["z", "2"]]]], RowBox[List["-", "n"]]], " ", SqrtBox[FractionBox[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"]], ")"]], "!"]]]], RowBox[List["4", " ", "\[Pi]"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["i", "!"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["k", "!"]], " ", SuperscriptBox["2", "i"], " ", SuperscriptBox["2", "j"]]]], RowBox[List["(", RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["i", "+", "j", "+", "k"]], ",", "n"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["i", "-", "j"]], ",", "m"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["x", "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]], "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "x"]], "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]], "i"], " ", SuperscriptBox["z", "k"]]], ")"]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "n"]], "\[LessEqual]", "m", "\[LessEqual]", "n"]], "\[And]", RowBox[List["x", "\[Equal]", RowBox[List[RowBox[List["Cos", "[", "\[CurlyPhi]", "]"]], RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]]]]]], "\[And]", RowBox[List["y", "\[Equal]", RowBox[List[RowBox[List["Sin", "[", "\[CurlyPhi]", "]"]], RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]]]]]], "\[And]", RowBox[List["z", "\[Equal]", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<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> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msup> <msqrt> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <mi> i </mi> <mo> + </mo> <mi> j </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> , </mo> <mi> n </mi> </mrow> </msub> <mo> ⁢ </mo> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <mi> i </mi> <mo> - </mo> <mi> j </mi> </mrow> <mo> , </mo> <mi> m </mi> </mrow> </msub> <mtext> </mtext> </mrow> <mrow> <mrow> <mi> i </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mi> i </mi> </msup> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mi> j </mi> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> x </mi> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> i </mi> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> ≤ </mo> <mi> m </mi> <mo> ≤ </mo> <mi> n </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> x </mi> <mo> ⩵ </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> φ </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> y </mi> <mo> ⩵ </mo> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> φ </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> z </mi> <mo> ⩵ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <ci> ϑ </ci> <ci> φ </ci> </apply> <apply> <times /> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> i </ci> <ci> j </ci> <ci> k </ci> </apply> <ci> n </ci> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> i </ci> </apply> <apply> <factorial /> <ci> j </ci> </apply> <apply> <factorial /> <ci> k </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <ci> i </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> <ci> j </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> <ci> i </ci> </apply> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> <apply> <leq /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> m </ci> <ci> n </ci> </apply> <apply> <eq /> <ci> x </ci> <apply> <times /> <apply> <cos /> <ci> φ </ci> </apply> <apply> <sin /> <ci> ϑ </ci> </apply> </apply> </apply> <apply> <eq /> <ci> y </ci> <apply> <times /> <apply> <sin /> <ci> φ </ci> </apply> <apply> <sin /> <ci> ϑ </ci> </apply> </apply> </apply> <apply> <eq /> <ci> z </ci> <apply> <cos /> <ci> ϑ </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", SuperscriptBox["z", "2"]]]], RowBox[List["-", "n"]]], " ", SqrtBox[FractionBox[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"]], ")"]], "!"]]]], RowBox[List["4", " ", "\[Pi]"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["i", "+", "j", "+", "k"]], ",", "n"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["i", "-", "j"]], ",", "m"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["x", "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]], "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "x"]], "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]], "i"], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["i", "!"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["k", "!"]], " ", SuperscriptBox["2", "i"], " ", SuperscriptBox["2", "j"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "n"]], "\[LessEqual]", "m", "\[LessEqual]", "n"]], "&&", RowBox[List["x", "\[Equal]", RowBox[List[RowBox[List["Cos", "[", "\[CurlyPhi]", "]"]], " ", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]]]]]], "&&", RowBox[List["y", "\[Equal]", RowBox[List[RowBox[List["Sin", "[", "\[CurlyPhi]", "]"]], " ", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]]]]]], "&&", RowBox[List["z", "\[Equal]", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
){kind=small}
