Chandrasekhar's X- and Y-function
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  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

Approximation
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)),  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

Properties
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.

See also
Chandrasekhar's H-function

References

Special functions
Integral equations
Scattering
Scattering, absorption and radiative transfer (optics)