Monday, 24 June 2013

Complex analysis on real integral



Use complex analysis to compute the real integral



dx(1+x2)3





I think I want to consider this as the real part of




dz(1+z2)3



and then apply the residue theorem. However, I am not sure how that is the complex form and the upper integral is the real part and how to apply.


Answer



Define a contour that is a semicircle in the upper half plane of radius R. Plus the real line from R to R



Then let R get to be arbitrarily large.



There is one pole at z=i inside the contour




Cauchy integral formula says:



f(n)(a)=n!2πif(a)(za)n+1 dz



1(z+i)3(zi)3 dz=πid2dz21(z+i)3 evaluated at z=i.



Next you will need to show that the integral along contour of the semi-cricle goes to 0 as R gets to be large.



z=Reit,dz=iReitπ0iReit(R2e2it+1)3 dt|iReit(R2e2it+1)3|<R5|π0iReit(R2e2it+1)3 dt|<π0R5 dtlim


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