Chandrasekhar's H-function


In atmospheric radiation, Chandrasekhar's H-function appears as the solutions of problems involving scattering, introduced by the Indian American astrophysicist Subrahmanyan Chandrasekhar. The Chandrasekhar's H-function defined in the interval, satisfies the following nonlinear integral equation
where the characteristic function is an even polynomial in satisfying the following condition
If the equality is satisfied in the above condition, it is called conservative case, otherwise non-conservative. Albedo is given by. An alternate form which would be more useful in calculating the H function numerically by iteration was derived by Chandrasekhar as,
In conservative case, the above equation reduces to

Approximation

The H function can be approximated up to an order as
where are the zeros of Legendre polynomials and are the positive, non vanishing roots of the associated characteristic equation
where are the quadrature weights given by

Explicit solution in the complex plane

In complex variable the H equation is
then for, a unique solution is given by
where the imaginary part of the function can vanish if is real i.e.,. Then we have
The above solution is unique and bounded in the interval for conservative cases. In non-conservative cases, if the equation admits the roots, then there is a further solution given by

Properties

and