Application of the Theory of Linear Singular Integral Equations to a Riemann Hilbert Problem for a New Expression of Chandrasekhar's H- function in Radiative Transfer

The linear non homogeneous singular integral equation (LNSIE)derived from the nonlinear non homogeneous integral eauation (NNIE)of Chandrsasekhar's H- functions is considered here to develop a new form of H - functions.The Plemelj's formulae are applied to that equation to determine a new linear non homogeneous integral equation(LNIE)for H- functions in complex plane . The analytic properties of this new linear integral equation are assessed and compared with known linear integral equations satisfied by H- functions. The Cauchy integral formulae in complex plane are used to obtain this form of H- functions not dependent on H- function in the integral . This new form of H-function is represented as a simple integral in terms of known functions both for conservative and non conservative cases. This is identical with the form of H- functions derived by this author by application of Wiener HOpf technique. The equivalence of application of the theory of linear singular integral equation in Riemann Hilbert Problem and of the technique of Wiener- Hopf in linear integral in representing the H- functions is therefore eatablished .This new form may be used for solving the problems of radiative transfer in anisotropic and non coherent scattering by the method of Laplace Transform and Wiener -Hopf technique.