In atmospheric radiation, Chandrasekhar's X- and Y-function appears as the solutions of problems involving diffusive reflection and transmission, introduced by the Indian American astrophysicist Subrahmanyan Chandrasekhar.[1][2][3][4][5] The Chandrasekhar's X- and Y-function defined in the interval , satisfies the pair of nonlinear integral equations
where the characteristic function is an even polynomial in generally satisfying the condition
and is the optical thickness of the atmosphere. If the equality is satisfied in the above condition, it is called conservative case, otherwise non-conservative. These functions are related to Chandrasekhar's H-function as
and also
The and can be approximated up to nth order as
where and are two basic polynomials of order n (Refer Chandrasekhar chapter VIII equation (97)[6]), where are the zeros of Legendre polynomials and , where are the positive, non vanishing roots of the associated characteristic equation
where are the quadrature weights given by