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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. The Chandrasekhar's X- and Y-function X , Y {\displaystyle X,\ Y} defined in the interval 0 ≤ μ ≤ 1 {\displaystyle 0\leq \mu \leq 1} , satisfies the pair of nonlinear integral equations
where the characteristic function Ψ {\displaystyle \Psi } is an even polynomial in μ {\displaystyle \mu } generally satisfying the condition
and 0 < τ 1 < ∞ {\displaystyle 0<\tau _{1}<\infty } 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