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The scattering and diffraction of a complex-source beam (CSB) by an acoustically soft or hard plane angular sector is treated by a rigorous spherical-multipole analysis in sphero-conal coordinates. By assigning a complex-valued radial source coordinate to the corresponding Green's function, the CSB is directed exactly towards the corner of the sector. Since the CSB can be interpreted as a localized plane wave, its interaction with the corner in the presence of the semi-infinite structure can be analyzed in detail. In opposite to the classical case of a non-localized incident plane wave, the resulting multipole series is strongly convergent and no summation techniques are necessary to obtain meaningful results. The numerical results include a convergence analysis, total near fields as well as scattered far fields and prove the applicability of this new approach.