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- #1

$$

u(r,\theta,\phi)\simeq 25 + \frac{75\sqrt{2}}{2}r\sin\theta\cos\phi + \frac{125}{\pi}r^2\sin^2\theta\cos 2\phi.

$$

How do I combine them?

The coefficients are

For $\ell = 0$ and $m = 0$, we have

$$

Y^0_0(\theta,\varphi) = \frac{1}{2\sqrt{\pi}}.

$$

For $\ell = 1$ and $m = 1$, we have

\begin{alignat*}{3}

Y^{1}_1 & = & -\frac{1}{2}\sqrt{\frac{3}{2\pi}}(\cos^2\theta - 1)^{1/2}e^{i\varphi}\\

& = & -\frac{1}{2}\sqrt{\frac{3}{2\pi}}e^{i\varphi} \sin\theta

\end{alignat*}

From $\ell = 1$ and $m = 1$, we can obtain

$$

Y^{-1}_1 = \frac{1}{2}\sqrt{\frac{3}{2\pi}}e^{-i\varphi}\sin \theta.

$$

For $\ell = 1$ and $m = 0$, we have

$$

Y^0_1 = \frac{\sqrt{3}}{2\sqrt{\pi}}\cos\theta.

$$

For $\ell = 2$ and $m = 2$, we have

\begin{alignat*}{3}

Y^{2}_2 & = & \frac{3}{4}\sqrt{\frac{5}{6\pi}}(\cos^2\theta - 1)e^{2i\varphi}\\

& = & \frac{3}{4}\sqrt{\frac{5}{6\pi}}e^{2i\varphi}\sin^2\theta

\end{alignat*}

From $\ell = 2$ and $m = 2$, we can obtain

\begin{alignat*}{3}

Y^{-2}_2 & = & \frac{3}{4}\sqrt{\frac{5}{6\pi}}e^{-2i\varphi}\sin^2 \theta.

\end{alignat*}

For $\ell = 2$ and $m = 1$, we have

\begin{alignat*}{3}

Y^{1}_2 & = & -\frac{3}{2}\sqrt{\frac{5}{6\pi}}e^{i\varphi}(\cos^2\theta - 1)^{1/2}\cos\theta\\

& = & -\frac{3}{2}\sqrt{\frac{5}{6\pi}}e^{i\varphi}\sin \theta \cos\theta

\end{alignat*}

From $\ell = 2$ and $m = 1$, we can obtain

\begin{alignat*}{3}

Y^{-1}_2 & = & \frac{3}{2}\sqrt{\frac{5}{6\pi}}e^{-i\varphi}\sin \theta \cos\theta.

\end{alignat*}

For $\ell = 2$ and $m = 0$, we have

$$

Y^0_{\ell} = \frac{1}{4}\sqrt{\frac{5}{2\pi}}(3\cos^2\theta - 1).

$$