$\mathbb R$-linear conjugation problem for a confocal parabolic annulus

The article presents an analytical closed-form solution derived for an $\mathbb R$-linear conjugation problem for a confocal parabolic annulus. The cases of real and complex coefficients of boundary conditions are comprehensively studied. The solution is found in the class of piece-wise holomorphic functions with fixed finite value at infinity in one of the media components.