Contour integration exp(ikx)/x

[aranud]

Homework Statement

Calculate the principal value integral exp(ikx)/x from -infinity to infinity first with a formula derived in the textbook and then by displacing the pole. Use this result to calculate the integral of sin(x)/x from -infinity to infinity.

Homework Equations

If the integral around the contour in the upper half of the plane goes to zero when the radius goes to infinity one can use the following formula:
integral f(x)/(x-x0) = i*pi/f(x0)+2pi*(residus poles) The poles in the domain around which you integrate.

The Attempt at a Solution

I've managed to prove with jordan's lemma that the upper part of the contour goes to zero when integrated so the formula can be used. Since the only pole is on the contour at x=0 we get that the integral is i*pi*1. This isn't so weird because the integrand isn't real either.

My problem is with the displacement of the pole technique. If we displace the pole north by changing the integrand to exp(ikx)/(x+iε) the integrand is split in an imaginary and real part:
limε→0 RE[int(sin(kx)/(x+iε)] + i limε→0 RE[int(icos(kx)/(x+iε))]

But because there are no poles in the domain around which I integrate these should both be zero? I think I'm missing something very basic here but Ican't figure out what it is.

Thanks.

 

[vela]

The way you close the contour should depend on the sign of k. Did you take that into account?

 

[aranud]

Well the problem statement just said k was real number so I tacitly assumed it to be positive and closed the contour in the half plane. The integral shouldn't depend on which half-plane you use to close the contour anyway.
edit: apparently the sign of the integral does depend on k. But if I manage to solve it for k positive I'll understand the k negative problem too.

 

[clamtrox]

Well the problem statement just said k was real number so I tacitly assumed it to be positive and closed the contour in the half plane. The integral shouldn't depend on which half-plane you use to close the contour anyway.
edit: apparently the sign of the integral does depend on k. But if I manage to solve it for k positive I'll understand the k negative problem too.

For the first calculation, you can just change your variables so that the exponential e^ikx is replaced with e^i|k|x, and you get your result easily. But what happens if you try doing after you have displaced the pole?