= Solution
For the <harmonic extension in a circular sector>, write a separated <harmonic function> as $F(r)H(\theta)$. The <Laplace equation in polar coordinates> yields
$$
\frac{r(rF')'}{F}=-\frac{H''}{H}=\nu^2.
$$
The angular <Dirichlet boundary conditions> select the <Fourier sine basis>
$$
H_n(\theta)=\sin\frac{n\theta}{2\alpha},\qquad \nu_n=\frac{n}{2\alpha},\quad n=1,2,\ldots.
$$
Indeed, the angular equation has no nonzero solution with nonpositive $\nu^2$: multiplying $-H''=\nu^2H$ by $H$ and integrating gives $\int(H')^2=\nu^2\int H^2$. The radial <Cauchy-Euler equation> has solutions $r^{\nu_n}$ and $r^{-\nu_n}$. Boundedness at the vertex excludes the latter, so <superposition> gives
$$
\boxed{\Phi(r,\theta)=\sum_{n=1}^\infty A_n r^{n/(2\alpha)}\sin\frac{n\theta}{2\alpha}.}
$$
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