Regular singular Volterra equations on complex domains

TL;DR

By converting linear differential equations on complex domains with irregular singularities into Volterra integral equations on the real domain via the inverse Laplace transform $\mathcal{L}$ defined by $\mathcal{L}\varphi := \int_{\Gamma_\zeta} e^{-z\zeta} \varphi\, d\zeta$, with $\Gamma_\zeta$ the ray $\zeta \in (0,\infty)$, the paper develops tailored Laplace-transform methods. For a class of regular-singular Volterra equations on complex domains, it proves existence and uniqueness of a solution of a prescribed form. The motivating level-$1$ ODE example demonstrates that the Laplace approach reproduces analytic solutions consistent with resummation (e.g., Borel or Stokes phenomena). This provides a direct analytic construction and clarifies the structure of such problems, expanding the applicability of Laplace-transform techniques to complex-domain Volterra problems with irregular singularities.

Abstract

The inverse Laplace transform can turn a linear differential equation on a complex domain into an equivalent Volterra integral equation on a real domain. This can make things simpler: for example, a differential equation with irregular singularities can become a Volterra equation with regular singularities. It can also reveal hidden structure, especially when the Volterra equation extends to a complex domain. Our main result is to show that for a certain kind of regular singular Volterra equation on a complex domain, there is always a unique solution of a certain form. As a motivating example, this kind of Volterra equation arises when using Laplace transform methods to solve a level 1 differential equation.