What is the value of $\int_{0}^{2\pi}f(e^{it}) \cos t \, dt$ if $f$ is analytic?

Question

If $f(z)$ is an analytic function, then find the value of the integration $$\int_{0}^{2\pi}f(e^{it}) \cos t \, dt.$$

My Work: Taking $e^{it} = z$ the integrand becomes of the form

$$i\frac{f(z)\operatorname{Re(z)}}{z}$$

on the simple closed contour $|z|=1$. But how can I proceed from here? Please help.

Answer 1

Hint: If you write $\cos t$ as $\frac {e^{it}+e^{-it}} 2$ you will see that the integral is $\frac 1 {2i}\int_{\gamma} \frac {f(z){(1+z^{2})}} {z^{2}}$. You can evaluate this using Cauchy's Integral Formula or the Residue Theorem.