The solution form you have written is for a disk not an annulus or half-annulus Separation of variables can be used in your half-annulus too, and the form you will get is$$A + B\ln r + \sum_{n=1}^{\infty} (A_n r^n \cos n\theta + B_n r^n \sin n\theta ) $$$$+ \sum_{n=1}^{\infty} (C_n r^{-n}\cos n\theta + D_n r^{-n} \sin n\theta )$$You then have to use the boundary values to find the $A_n, B_n, C_n,$ and $D_n$.
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