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
- If are the solutions for a particular value of , then solutions for other values of are obtained from the following integro-differential equations
- For conservative case, this integral property reduces to
- If the abbreviations for brevity are introduced, then we have a relation stating In the conservative, this reduces to
- If the characteristic function is , where are two constants, then we have .
- For conservative case, the solutions are not unique. If are solutions of the original equation, then so are these two functions , where is an arbitrary constant.
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