Let $f$ be an holomorphic function on $\mathbb C \setminus \{0\}$ and bounded on open set which contain the origine.
Question Evaluate the integral $$ \int_{|z|=5} f(z) dz $$ My proposition : with the prametrization $z(t)=5e^{i t}, \quad 0< t< 2 \pi$ \begin{align*} \int_{|z|=5} f(z) dz & = \int_0^{2 \pi} f(z(t))z'(t)dt \\ & = \int_0^{2 \pi} f(e^{i t})5 i e^{i t} dt \\ & = 5 i \int_0^{2 \pi} f(e^{i t}) e^{i t} dt \end{align*} As $f$ is bounded ( $|f(z)| \leq M$), then \begin{align*} \left|\int_{|z|=5} f(z) dz \right|& \leq \int_0^{2\pi}\left| f(z(t))z'(t)\right|dt \\ & =| 5 i |\int_0^{2 \pi}\left| f(e^{i t})\right| dt \\ & \leq 5 M \times 2 \pi. \end{align*} I can't conclude of the value of the integral.
Any help is welcome.