Cauchy Principal Value

Question

How do we evaluate the Cauchy Principal value for:

$$ \int_{-\infty}^\infty\frac{\cos kx}{x-a}dx $$

Given, a is real, k >${\ 0}$?

I thought of integrating from ${-\infty}$ to ${\ a}$ and then from ${\ a}$ to ${+\infty}$

Any help will be appreciated.

Answer 1

Change variables, $y=x-a$, so $\cos{kx} = \cos{(ky+ka)} = \cos{ky}\cos{ka}-\sin{ky}\sin{ka}$, so $$ I = \cos{ka}\int_{-\infty}^{\infty} \frac{\cos{ky}}{y} \, dy - \sin{ka} \int_{-\infty}^{\infty} \frac{\sin{ky}}{y} \, dy $$ The former is odd, so its principal value is zero (one could check this by considering the symmetrical limit at $y=0$ as usual). The latter integral is well-known to be $\pi\operatorname{sgn}{k}$, so the answer is $$ -\pi \sin{ka}\operatorname{sgn}{k} = -\pi\sin{(\lvert k \rvert a)}. $$