Friday, 10 May 2019

complex analysis - Show that intnolimitsinfty0x1sinxdx=fracpi2

Show that 0x1sinxdx=π2 by integrating z1eiz 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 ϵ0 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: z1eiz (Ref: http://www.roadsolutions.ox.ac.uk/corrections.html)

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