Yes (except that one uses almost a full circle rather than a semicircle, the contour is a “keyhole contour” https://en.m.wikipedia.org/wiki/File:Keyhole_contour.svg

]]>One uses a contour that loops around the *positive* real axis, e.g. the clockwise semicircle from to , together with the ray from to and the ray from $-\varepsilon i + i \infty$ to (cf. the “keyhole contours” that are often used in complex analysis).

I think using the product form of the expression in the limit is indeed the fastest way to show convergence.

]]>The main thing is to introduce the variable ; it is not really relevant what the remaining variable is chosen to be, as the integral will simplify regardless, basically because the exponential weight simplifies to .

]]>Consider the inverse of the map you take (using different notations):

It seems that this is related to some fractional linear transformation? Is there anything in complex analysis that would lead one naturally to the change of variables map used in the proof of Lemma 7?

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