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One I can't do

Many years ago someone told me that Littlewood found a way to use the Residue Theorem (or perhaps complex methods in general) to find $\int_{-\infty}^\infty e^{-t^2/2}\,dt$. I remain stumped - any clues?

One I can do

Teaching baby complex, it's unfortunate that the students get the idea that $\int_{-\infty}^\infty f(t)\,dt$ is sort of automatically $2\pi i$ times the sum of the residues in the upper half plane, without needing to worry about showing that the "other" part of the contour integral tends to $0$. I noticed a simple example regarding that:

Exercise. Use the Residue Theorem to find $\lim_{R\to\infty}\int_{-R}^R\frac{dt}{t+i}$.

It's a straightforward argument; turns out that the integral is half of $2\pi i$ times the sum of the residues in the upper half-plane...

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David C. Ullrich
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  • There is this, perhaps you have seen it - https://math.stackexchange.com/questions/34767/int-infty-infty-e-x2-dx-with-complex-analysis?noredirect=1&lq=1 – Calvin Khor Aug 28 '20 at 13:48
  • There is a PDF referenced in a comment on this question, and the PDF seems to have what you want. https://math.stackexchange.com/questions/390850/integrating-int-infty-0-e-x2-dx-using-feynmans-parametrization-trick . The PDF link is https://kconrad.math.uconn.edu/blurbs/analysis/gaussianintegral.pdf – Disintegrating By Parts Jun 26 '21 at 00:37

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