Factoring Legendre function?

[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!

 

[gtoguy97]

Yes, it is possible to factor the Legendre functions P_k(cos(theta-theta1)) out in separate functions of theta and theta1 in two dimensions. The result is a product of two Legendre polynomials, one depending on theta and the other on theta1. For example, the term P_k(cos(theta - theta1)) can be written as P_k(cos(theta-theta1)) = P_k(cos(theta))*P_k(cos(theta1)) where P_k(cos(theta)) and P_k(cos(theta1)) are the Legendre polynomials of degree k in theta and theta1, respectively.