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I want to expand the potential of this charge distribution using spherical coordinates.Here's how I did it:

[itex]

\phi=\frac {Q} {4\pi \varepsilon_0} \left[ \frac{1}{r} - \frac{1}{\sqrt{r^2+d^2-2rd \cos\theta}}- \frac{1}{\sqrt{r^2+d^2-2rd \cos\gamma_1}}- \frac{1}{\sqrt{r^2+d^2-2rd \cos\gamma_2}}\right]=\\

\frac {Q} {4\pi \varepsilon_0 r} \left[ 1 - \frac{1}{\sqrt{1+(\frac d r)^2-2 \frac d r \cos\theta}}- \frac{1}{\sqrt{1+(\frac d r)^2-2\frac d r \cos\gamma_1}}- \frac{1}{\sqrt{1+(\frac d r)^2-2\frac d r \cos\gamma_2}}\right]=\\

\frac {Q} {4\pi \varepsilon_0 r} \left[

1-\sum_{n=0}^\infty P_n(\cos\theta) (\frac d r)^n-\sum_{n=0}^\infty P_n(\cos\gamma_1) (\frac d r)^n-\sum_{n=0}^\infty P_n(\cos\gamma_2) (\frac d r)^n

\right]=\\

\frac {Q} {4\pi \varepsilon_0 r} \left[

1-\sum_{n=0}^\infty P_n(\cos\theta) (\frac d r)^n-\sum_{n=0}^\infty \frac{4\pi}{2n+1} (\frac d r)^n\sum_{m=-n}^n Y^{m*}_n(\frac \pi 2,0) Y^m_n(\theta,\varphi)-\sum_{n=0}^\infty \frac{4\pi}{2n+1} (\frac d r)^n\sum_{m=-n}^n Y^{m*}_n(\frac \pi 2,\frac \pi 2) Y^m_n(\theta,\varphi)\right]

[/itex]

The monopole term(n=0) is [itex] \phi^{(1)}=-\frac{2Q}{4\pi \varepsilon_0 r } [/itex],as it should be.

My problem is, the dipole term(n=1) turns out to be complex.What's wrong?

Thanks