The finest traditions of Russian applied mathematics are represented in this paper. It is also very formal; not only are the definitions numbered 1.1, 1.2, and so on, like the theorems, but the remarks and assertions are serially numbered as well.

I found the paper to be a long, challenging read, but well worthwhile. The author considers a linear ordinary differential equation (ODE) with a resonant irregular, but generic, singularity. He splits the irregular singularity of the unperturbed equation into Fuchsian singularities of the perturbed one, and then expresses its analytic invariants as limit transition operators that compare appropriate monodromy eigenbases of the perturbed equation. Finally, he shows that appropriate branches of the monodromy eigenfunctions of the perturbed equation converge to the corresponding canonical solutions of the unperturbed equation. The monodromy branching is well illustrated by graphics, and they make it much easier to follow the algebra, which gets a bit complicated in places.

Since many of the author’s references are to Russian language and other, not readily available, foreign mathematical journals, I found it helpful to read the paper with my laptop computer tuned to the Wikipedia mathematics section uniform resource locator (URL). If you are working in this same area of applied mathematics, you may not need this intellectual prosthesis.