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Spherical Harmonics: Difference between revisions
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==Background== | ==Background== | ||
===Harmonic Function=== | ===Harmonic Function=== | ||
{{main | Wikipedia: Harmonic_function}} | |||
A function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}</math> is a harmonic function if it satisfies Laplace's equation: | A function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}</math> is a harmonic function if it satisfies Laplace's equation: | ||
* The Laplacian (or trace of the hessian) is zero. | * The Laplacian (or trace of the hessian) is zero. | ||
* <math>\Delta f = \frac{\partial^2f}{\partial x_1^2} + \frac{\partial^2f}{\partial x_2^2} + \cdots + \frac{\partial^2f}{\partial x_n^2} = 0</math> | * <math>\Delta f = \frac{\partial^2f}{\partial x_1^2} + \frac{\partial^2f}{\partial x_2^2} + \cdots + \frac{\partial^2f}{\partial x_n^2} = 0</math> | ||
===Associated Legendre Polynomials=== | |||
{{main | Wikipedia:Associated Legendre polynomials}} | |||
Associated Legendre Polynomials are a set of orthogonal polynomials defined over \([-1, 1]\). | |||
The following 3 recurrance relations define the associated legendre polynomials: | |||
# <math>(l-m)P_l^m = x(2l-1)P_{l-1}^m - (l+1-1) P_{l-2}^m</math> | |||
# <math>P_m^m = (-1)^m(2m-1)!! (1-x^2)^{m/2}</math> | |||
# <math>P^m_{m+1} = x(2m+1)P^m_m</math> | |||
Notes: | |||
* Here <math> | |||
(x)!! = | |||
\begin{cases} | |||
(x)*(x-2)*...*(1) & x\text{ odd}\\ | |||
(x)*(x-2)*...*(2) & x\text{ even} | |||
\end{cases} | |||
</math> | |||
==Definition== | ==Definition== | ||