The finite Laplace transform for solving a weakly singular integral equation occurring in transfer theory

We solve a weakly singular integral equation by Laplace transformation over a finite interval of R. The equation is transformed into a Cauchy integral equation, whose resolution amounts to solving two Fredholm integral equations of the second kind with regular kernels. This classical scheme is used to clarify the emergence of the auxiliary functions expressing the solution of the problem. There are four such functions, two of them being classical ones. This problem is encountered while studying the propagation of light in strongly scattering media such as stellar atmospheres.