# Being precise about contour integrals when dealing with branch-cuts and singularities

In this paper https://www.biodiversitylibrary.org/item/93357#page/65/mode/1up of Landau he wants to evaluate $$\int _{2-iT}^{2+iT}\frac {x^s\log (s-1)}{s^2}\hspace {10mm}(1)$$ (page 755). He sets up a keyhole-type contour path, something like $$\int _{\text {path from 2+iT to 1/2}}+\int _{L_1}+\int _{L_2}+\int _{\text {path from 1/2 to 2-iT}}\hspace {10mm}(2)$$ where $$L_1,L_2$$ are paths between 1/2 and 1, essentially on the real line. He then applies Cauchy's Residue Theorem - he says (1) and (2) are equal.

I feel I understand the general idea, and understand that we choose a different argument for $$\log (s-1)$$ when we take the path the second time, and that it's this $$2\pi i$$ increment that gives us a main term. (I also understand that each individual integral exists because $$\log$$ growth means we can take the integrals up to the singular point.) Still, when I'm pushed to write down concretely and precisely what's happening then I find I can't do it.

Specifically:

1.Cauchy's Integral Theorem, as far as I know, is a statement about functions on a subset of $$\mathbb C$$. In that case, what exactly is the subset we take above? Is it $$\mathbb C-\mathbb R^{\leq 0}$$ for example? It could well be that I'm oversimplifying and that it's not just a subset.

1. What precisely are the paths $$L_1$$ and $$L_2$$? Are they literally on the real axis or are they just above and below, with an $$\epsilon$$ argument that's not written down explicitly?

2. Does the path self-intersect along the real axis, and if so how do we apply Cauchy's Theorem?

I'd like it if someone could tell me precisely where/how Cauchy's Integral Theorem is being applied, perhaps having to be very careful about the exact statement.

Sorry for being so pedantic it just seems to be something I need to get clear! Thanks:)

• They are two paths approximating the real axis from above and below. This treatment is quite common in the theory of special functions back then. Nov 20 at 21:55
• what does "approximating" mean? can you write down literally what those paths are? (again sorry for being insisting here:D) Nov 21 at 4:38

When $$\log$$ is meant to take the principal value, the function $$s^{-2}x^s\log(s-1)$$ is holomorphic and one-valued inside and on the contour described as follows.
Consequently, Cauchy's theorem applies, and notice that the integral over the circle around $$s=1$$ vanishes when the radius of the circle tends to zero, so you will obtain the desired result.