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$$

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?

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?

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