Calculating the principal value of an Integral

[Demon117]

Homework Statement

$P.v.\int e^{itx}/(x^{2}-k^{2})dx$

Where, $t,k\in Reals$ and the range of integration is $-\infty$ to $\infty$.

Homework Equations

Not sure.

The Attempt at a Solution

I tried to find a solution to this problem by looking at the residues in each plane, but it seems that this is incorrect. Does one just take the real part/imaginary part of the integral? This is something I don't quite understand and I have no experience with. Could someone please give me some pointers/advice on these types of problems?

 

[mighty2000]

Hi there, it seems like you are on the right track with looking at the residues in each plane. However, in order to find the correct solution, you will also need to use the Cauchy Residue Theorem. This theorem states that if f(z) is a complex-valued function that is analytic everywhere inside and on a simple closed contour C, and has an isolated singularity at z0 inside C, then the integral of f(z) around C is equal to 2πi times the residue of f(z) at z0. In this case, you will need to find the residues at the poles of the integrand, which are at x=k and x=-k. You can then use the Cauchy Residue Theorem to evaluate the integral. I hope this helps! Let me know if you have any further questions.