Complex integral on a circumference

Question

I have to integrate the function $$f(z) = \frac{1}{1+z^2} e^{-2\pi i z \xi}$$ on the upper part of the circumference with center $0$ and radius $R$ in the complex plane. $\xi$ is a real number. I tried to use polar coordinates but it doesn't help. Can you give me a hint?

Answer 1

For this integral, you might consider using the so-called ML inequality, which states that if $ |f(z)| \leq M$ and the length of the contour $\gamma$ is less than $L$, then $$ \left|\int_\gamma f(z) dz\right| \leq ML.$$

Back to the question, can you try to compute the bound of $f$ on the semi-circle on the upper plane? Try to substitute $z = Re^{i\theta}$ directly and simplify the exponent.