Eigenvalues of holomorphic functions for the third boundary condition

The eigenvalue problem of holomorphic functions on the unit disc for the third boundary condition with general coefficient is studied using Fourier analysis. With a general anti-polynomial coefficient a variable number of additional boundary conditions need to be imposed to determine the eigenvalue uniquely. An additional boundary condition is required to obtain a unique eigenvalue when the coefficient includes an essential singularity rather than a pole. In either case explicit solutions are derived.