Summary: In radiative transfer, the intensities of radiation from the bounding faces of a scattering atmosphere of finite optical thickness can be expressed in terms of Chandrasekhar’s \(X\)- and \(Y\)-functions. The nonlinear nonhomogeneous coupled integral equations which the \(X\)- and \(Y\)-functions satisfy in the real plane are meromorphically extended to the complex plane to frame linear nonhomogeneous coupled singular integral equations. These singular integral equations are then transformed into nonhomogeneous Riemann-Hilbert problems using Plemelj’s formulae. Solutions of those Riemann-Hilbert problems are obtained using the theory of linear singular integral equations. New forms of linear nonhomogeneous decoupled expressions are derived for \(X\)- and \(Y\)-functions in the complex plane and real plane. Solutions of these two expressions are obtained in terms of one known \(N\)-function and two new unknown functions \(N_{1}\)- and \(N_{2}\)- in the complex plane for both nonconservative and conservative cases. The \(N_{1}\)- and \(N_{2}\)-functions are expressed in terms of the known \(N\)-function using the theory of contour integration. The unknown constants are derived from the solutions of Fredholm integral equations of the second kind uniquely using the new linear decoupled constraints. The expressions for the \(H\)-function for a semi-infinite atmosphere are obtained as a limiting case.