Show that ∫∞0x−1sinxdx=π2 by integrating z−1eiz around a closed contour Γ consisting of two portions of the real axis, from -R to -ϵ and from ϵ to R (with R>ϵ>0) and two connecting semi-circular arcs in the upper half-plane, of respective radii ϵ and R. Then let ϵ→0 and R→∞.
[Ref: R. Penrose, The Road to Reality: a complete guide to the laws of the universe (Vintage, 2005): Chap. 7, Prob. [7.5] (p. 129)]
Note: Marked as "Not to be taken lightly", (i.e. very hard!)
Update: correction: z−1eiz (Ref: http://www.roadsolutions.ox.ac.uk/corrections.html)
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