Why Does Mathematica Give a Different Result for Spherical Harmonics?

[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?

 

[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.

 

[Dustinsfl]

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.

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