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.^{[1][2][3][4][5]} The Chandrasekhar's X- and Y-function ${\displaystyle X(\mu ),\ Y(\mu )}$ defined in the interval ${\displaystyle 0\leq \mu \leq 1}$, satisfies the pair of nonlinear integral equations

${\displaystyle {\begin{aligned}X(\mu )&=1+\mu \int _{0}^{1}{\frac {\Psi (\mu ')}{\mu +\mu '}}[X(\mu )X(\mu ')-Y(\mu )Y(\mu ')]\,d\mu ',\\[5pt]Y(\mu )&=e^{-\tau _{1}/\mu }+\mu \int _{0}^{1}{\frac {\Psi (\mu ')}{\mu -\mu '}}[Y(\mu )X(\mu ')-X(\mu )Y(\mu ')]\,d\mu '\end{aligned}}}$

where the characteristic function ${\displaystyle \Psi (\mu )}$ is an even polynomial in ${\displaystyle \mu }$ generally satisfying the condition

${\displaystyle \int _{0}^{1}\Psi (\mu )\,d\mu \leq {\frac {1}{2}},}$

and ${\displaystyle 0<\tau _{1}<\infty }$ 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

${\displaystyle X(\mu )\rightarrow H(\mu ),\quad Y(\mu )\rightarrow 0\ {\text{as}}\ \tau _{1}\rightarrow \infty }$

and also

${\displaystyle X(\mu )\rightarrow 1,\quad Y(\mu )\rightarrow e^{-\tau _{1}/\mu }\ {\text{as}}\ \tau _{1}\rightarrow 0.}$