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.

[1][2][3][4][5] The Chandrasekhar's H-function

defined in the interval

, satisfies the following nonlinear integral equation where the characteristic function

satisfying the following condition If the equality is satisfied in the above condition, it is called conservative case, otherwise non-conservative.

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 The H function can be approximated up to an order

are the zeros of Legendre polynomials

are the positive, non vanishing roots of the associated characteristic equation where

are the quadrature weights given by In complex variable

, a unique solution is given by where the imaginary part of the function

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