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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:)

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  • $\begingroup$ 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. $\endgroup$
    – TravorLZH
    Nov 20 at 21:55
  • $\begingroup$ what does "approximating" mean? can you write down literally what those paths are? (again sorry for being insisting here:D) $\endgroup$
    – tomos
    Nov 21 at 4:38

1 Answer 1

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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.

Contour illustration

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.

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  • $\begingroup$ i get the idea, but can you write down for me precisely what the application of cauchy's theorem is? on what domain exactly, and what path exactly? (starting with your black contour as 1 and going round, i'm not worried about contours 1,2,3,7,8. but i'd like to see a concrete definition of contour 4 for example) $\endgroup$
    – tomos
    Nov 21 at 4:37
  • $\begingroup$ i think i always assumed cauchy was applied to simply-connected. but we can apply it to any open set provided the curve shrinks to a point right? and that's the version we're applying? i think that was my problem $\endgroup$
    – tomos
    Nov 21 at 8:13
  • $\begingroup$ The cut plane is simply connected. $\endgroup$
    – TravorLZH
    Nov 21 at 12:19
  • $\begingroup$ so those paths are not on the real line? $\endgroup$
    – tomos
    Nov 21 at 16:20
  • $\begingroup$ They are not meant to be on the real line. They lie on the real line only after you take limits. $\endgroup$
    – TravorLZH
    Nov 22 at 23:46

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