# Factoring Legendre function?

1. Jul 20, 2010

### aamir.ahmed

Consider the multipole expansion of Newtonian potential 1/R in 2D in terms of Legendre functions.

\begin{align} \mathbf{r_1} &= (r , \theta_1) \\ \mathbf{r} &= (r , \theta) \\ \phi(\mathbf{r}-\mathbf{r_1}) &= \frac{1}{\left|\mathbf{r}-\mathbf{r_1}\right|} \\ \phi(\mathbf{r}-\mathbf{r_1}) &= \frac{1}{\sqrt{r^2+r_1^2 - 2 r r_1 cos(\theta - \theta_1) }} \\ \phi(\mathbf{r}-\mathbf{r_1}) &= \sum_{k=0}^{\infty} P_k\left(cos(\theta - \theta_1)\right) \frac{r_1^k}{r^{k+1}} \end{align}

I am working on an N-body simulation problem and would like to know if there is a way to factor the Legendre functions P_k(cos(theta-theta1)) out in separate functions of theta and theta1? Actually this is possible when using spherical harmonics in 3D, but I need it in 2D. Please help!

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