For certain problems, taking the Borel sum of an intuitively chosen divergent series solution will often produce a holomorphic solution. To understand when this happens, and what's special about the holomorphic solutions we get this way, it can be helpful to think of Borel summation as part of a regularization process acting on holomorphic functions. From this perspective, the problems Borel summation can solve and the solutions it produces are both characterized by regularity properties. We'll see how this story plays out for a linear ODE problem: finding a basis of holomorphic solutions for each of the Airy-Lucas equations, including the Airy equation as a special case.