# Looking for some interesting complex integration contours

I am currently working on some tools to make contour integration in a proof assistant less painful and I'm looking for interesting examples of contours in the complex plane used in the literature. I am most interested in contours that add little epsilon detours in order to avoid singularities/branch cuts/etc.

One example of what I am looking for is the following contour that is used in the study of modular functions.

[An interesting integration contour from Apostol's "Modular Functions and Dirichlet Series in Number Theory"]

Here are a few contours of diverse complexity (click on the image for the link to the source publication).

• It's a start, thanks! But I'm much more interested in examples where some relatively simple path is distorted by some epsilon to avoid a singularity on the path, as in my example. The second example looks like it could be that? The first one not so much. But in any case, thanks! I'll have a look at them, but I'm looking for as many different examples as possible. Mar 26, 2022 at 12:22
• added two more, the fourth one looks simple enough? Mar 26, 2022 at 12:59
• Great, thanks for that! Mar 26, 2022 at 13:15
• It looks like some biological scheme (yes, pun intended). Mar 28, 2022 at 8:21

At a very general level, an autonomous evolution differential equation writes $$\frac{du}{dt}=Lu,$$ where $$L$$, an unbounded linear operator, can be rather complicated. For instance, in parabolic systems of conservation laws, we encouter 2nd-order differential (in space variables) operator with variable coefficients, and $$u=(u_1,\ldots,u_m)$$ is vector valued. The differential system is then completed with boundary conditions, which may be more complicated than just Dirichlet or Neuman.

Typically, the situation above arises when studying the stability of steady solutions of non-linear problem. This study is split into three intermediate steps:

1. Spectral stability: is the spectrum of $$L$$ included in the left half of the complex plane ?
2. Linear stability: if so, is the semi-group $$S_t=e^{tL}$$ bounded ? What dissipation inequality does it display ?
3. From linear to nonlinear stability. Here one uses the Duhamel formula and a fixed point argument.

Contours of integration arise in the second step above. One expresses $$S_t$$ as a Cauchy integral, in terms of the resolvent $$R(z)=(z I-L)^{-1}$$, the latter being well-defined and holomorphic over the resolvent set $$\rho(L)$$. One integrates over a contour $$\Gamma\subset\rho(L)$$. In practice, the branches at infinity stay away (from the left) of the imaginary axis ; it is ensured by a property of the essential spectrum.

One typical difficulty is that the steady solution $$\phi$$ is often not isolated, because it can be translated in space. This implies that $$z=0$$ belongs to the spectrum. For instance, in one space dimension, we have $$L\frac{d\phi}{dx}=0$$. Thus the contour must pass somewhere in the right half-plane, to avoid $$z=0$$. But of course, establishing good estimates for $$S_t$$ requires that $$\Gamma$$ be $$\epsilon$$-close to $$z=0$$.

As you can see, this is a general philosophy. The details vary according to the differential equation under consideration. It was initiated by D. H. Sattinger in 1977 and then considerably developped until recent times by many authors. It is a very active research topic.

"I'm interested in examples where a relatively simple path is distorted by some epsilon to avoid a singularity": The proof of Theorem 5.2 in Montgomery and Vaughan's "Multiplicative Number Theory" seems to fit. Just a rectangle and a detour around zero. (This is Perron's formula, and proofs without the detour is possible at the cost of a less explicit error term).

The book Balades sur les chemins du champ complexe of Gilbert Demengel offers a wide variety of contours (of varying complexity) that should be quite relevant to your project.

See page 382 in [1], a paper I have found very enlightening.

Reference

[1] HARDY, G. H. (1905), "A method for determining the behavior of certain classes of power series near a singular point on the circle of convergence", Proceedings of the London Mathematical Society 2, pp. 381-389, JFM 36.0474.01.