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!2πi∮f(a)(z−a)n+1 dz
∮1(z+i)3(z−i)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=iReit∫π0iReit(R2e2it+1)3 dt|iReit(R2e2it+1)3|<R−5|∫π0iReit(R2e2it+1)3 dt|<∫π0R−5 dtlim
No comments:
Post a Comment