The time-dependent \(X\)- and \(Y\)-functions (Q1201768)

Time-dependent \(X\)- and \(Y\)-functions are considered which are analogous to the \(X\)- and \(Y\)-functions studied by \textit{S. Chandrasekhar} [Radiative transfer (1960)] in relation with the theory of radiative transfer. These functions satisfy the following coupled nonlinear integral equations \[ \begin{aligned} X(\tau,\mu,s) &= 1+{W \over 2Q}\mu \int_ 0^ 1 {{X(\tau_ 1,\mu,s)X(\tau_ 1,x,s)-Y(\tau_ 1,\mu,s)Y(\tau_ 1,x,s)} \over {\mu+x}}dx,\\ Y(\tau_ 1,\mu,s) &= \exp\bigl(- {\tau_ 1Q \over \mu}\bigr) +{W \over 2Q}\mu\int_ 0^ 1 {{Y(\tau_ 1,\mu,s)X(\tau_ 1,x,s)X(\tau_ 1,\mu,s)Y(\tau_ 1,x,s)} \over {\mu- x}}dx,\end{aligned} \] \(Q=1+s/c\), \(0\leq\mu<1\). Using the Wiener-Hopf technique the latter are reduced to the pair of Fredholm integral equations, which define the time-dependent \(X\)-functions in terms of the time-dependent \(Y\)-functions and vice versa. These representations are unique with respect to the coupled linear constraints defined by \textit{T. W. Mullikin} [Trans. Am. Math. Soc. 113, 316-332 (1964; Zbl 0124.059)].