# Coefficient spherical harmonics

1. Oct 31, 2012

### Dustinsfl

$$Y_{\ell}^m = \sqrt{\frac{(2\ell + 1)(\ell - m)!}{4\pi(\ell + m)!}}P^m_{\ell}(\cos\theta)e^{im\varphi}$$

For $\ell = m = 1$, we have
$$\sqrt{\frac{(2 + 1)(0)!}{4\pi(2)!}}P^1_{1}(\cos\theta)e^{i\varphi} = \frac{1}{2}\sqrt{\frac{3}{2\pi}}e^{i\varphi}\sin \theta$$

But Mathematica is telling me the solution is
$$-\frac{1}{2} e^{i\varphi} \sqrt{\frac{3}{2\pi}} \sin\theta$$

What is going wrong?

2. Oct 31, 2012

### dextercioby

Have you checked the definitions/conventions for the associated Legendre functions ? The definitions for Mathematica you can find on the functions.wolfram.com website. Unfortunately, it's difficult to say that special functions theory is a unitary process with unique definitions.

3. Oct 31, 2012

### Dustinsfl

Some are defined with a (-1)^m which is weird my professor was using that definition since we were not in class.