The singularly perturbed Riccati equation is one of the simplest first-order nonlinear ODEs of the form $\hbar f' = af^2 + bf + c$ containing a small perturbation parameter $\hbar$. This equation plays an absolutely crucial role in the exact WKB analysis of Schrödinger equations and more general families of meromorphic connections on Riemann surfaces, where the goal is to canonically construct analytic $\hbar$-dependent data (e.g., solutions, flat sections, filtrations of vector bundles, etc) out of the corresponding asymptotic data obtained in the limit as $\hbar \to 0$. Part of the appeal is that the latter is normally a lot easier to analyse, much in the same way as a power series is usually a simpler object than the corresponding analytic function. However, the trouble is that attempting to solve the Riccati equation in power series in $\hbar$ almost always leads to divergent series, and converting this formal data into the desired analytic data (i.e., promoting formal solutions to exact solutions) is quite a nontrivial task. This is a quintessential problem in singular perturbation theory.

I will carefully go through the proof of an existence and uniqueness theorem for exact solutions from my recent paper (arXiv:2008.06492). The uniqueness part of this theorem is particularly important because it means that our construction gives a canonical way to promote formal solutions (i.e., asymptotic data) to exact solutions (i.e., analytic data). The construction is via the Borel-Laplace method (a.k.a., Borel resummation). I will therefore give a crash course in the theory of Borel-Laplace transforms and Gevrey asymptotics, all the while keeping the discussion as elementary as possible.