  In Radiative transfer, the intensities of radiation from the bounding faces
of finite slab are obtained in terms of X- and Y- functions of Chandrasekhar .
Those are non linear non homogeneous coupled integral equations . Those non
linear integral equations are meromorphically extended to the complex plane to
get linear non homogeneous coupled integral equations. Those linear integral
equations are converted to linear singular integral equations with. linear
constraints . Those singular integral equations are then transformed to non
homogeneous Riemann Hilbert Problems. Solutions of those Riemann Hilbert
Problems are obtained using the theory of linear singular integral equations to
decouple those X- and Y- functions. New forms of linear non homogeneous
decoupled integral equations are derived for X- and Y- function separately with
new linear constraints. Those new decoupled integral equations are transformed
into linear singular integral equations to get two new separate non homogeneous
Riemann Hilbert problems and to find solutions in terms of one known N-
function and five new unknown functions in complex plane . All five functions
are represented in terms of N-functions using the theory of contour integration
>. Those X- and Y- functions are finally expressed in terms of that N -
function and also in terms of H- functions of Chandrasekhar and of integrals in
Cauchy principal value sense in the complex plane and real plane. both for
conservative and non conservative cases . The H - functions for semi infinite
atmosphere are derived as a limiting case from the expression of X- function of
finite atmosphere.
