Chandrasekhar's X- and Y-functions

Example

Let $\tau_1 > 0$ be the optical thickness of a plane-parallel slab. Let $\Psi$ be an even polynomial, the characteristic function, with $\Psi \ge 0$ on $[0, 1]$ and $\int_0^1 \Psi\, d\mu \le \frac{1}{2}$. Chandrasekhar's X- and Y-functions are the functions $X$ and $Y$ on $[0, 1]$ that solve the system $$X(\mu) = 1 + \mu \int_0^1 \frac{\Psi(\mu')}{\mu + \mu'} \bigl[X(\mu) X(\mu') - Y(\mu) Y(\mu')\bigr]\, d\mu',$$ $$Y(\mu) = e^{-\tau_1/\mu} + \mu \int_0^1 \frac{\Psi(\mu')}{\mu - \mu'} \bigl[Y(\mu) X(\mu') - X(\mu) Y(\mu')\bigr]\, d\mu'.$$ In the second equation the bracket vanishes at $\mu' = \mu$, so the integrand has no singularity there. This is a system of two coupled nonlinear integral equations for the two unknowns.

Check of two limits. For $\tau_1 = 0$, the constant functions $X = 1$ and $Y = 1$ solve the system: both brackets vanish and $e^{0} = 1$. This is a slab of thickness $0$, which lets all light through. As $\tau_1 \to \infty$, the term $e^{-\tau_1/\mu}$ tends to $0$; with $Y = 0$ the first equation becomes the equation of Chandrasekhar's H-function, $X = H$.