# Spherical Harmonics Change of Coordinate System

## Homework Statement

Let $$\vec H = ih_4^{(1)}(kr)\vec X_{40}(\theta,\phi)\cos(\omega t)$$
where $h$ is Hankel function of the first kind and $\vec X$ the vector spherical harmonic.
a) Find the electric field in the area without charges;
b) Find both fields in a spherical coordinate system that is a rotation of 45 deg about the y axis.

## Homework Equations

Maxwell's equations, addition theorem for spherical harmonics.

## The Attempt at a Solution

For part a I used Maxwell's equations, namely
$$\nabla \times H = -\epsilon_0\frac{\partial E}{\partial t}$$
For part b I want to use the addition theorem, namely
$$P_l(\cos \alpha) = \frac{4\pi}{2l+1}\sum_{m=-l}^lY^{*}_{lm}(\theta ',\phi ')Y_{lm}(\theta,\phi)$$
using the specific transofrmation $\theta = pi/4,\;\phi=0$, but I can't find a way to isolate $Y_{lm}$ in the old coordinate system and express it in the new because of the terms in the sum. Any directions? Thanks.