Let $f :\Bbb C \to \Bbb C$ be an entire function.
Compute $\int_{0}^{2π} $$f$($re^{iθ}$) $dθ$, $r > 0$.
MY ATTEMPTS : From Cauchy's integral formula,
$$
\lvert f^{(n)}(0) \rvert \le \frac{n!}{2 \pi r^n} \int_0^{2\pi}
\lvert f(re^{i\theta}) \rvert \, d\theta
$$
Now, I don't know how to to proceed further.
Please help me.
Put $z=re^{i\theta}$.
Then $dz=izd\theta$.
Thus the integral becomes $\frac 1i \int_{\gamma} \frac {f(z)}{z} dz$ which is equal to $2 \pi f(0)$ by Cauchy integral formula where $\gamma$ is a circle $|z|=r$.