Meaning of $d \theta$

Question

I am confused with the notation $d \theta$. Given a function $f: \Bbb C \to \Bbb C$, is $\int _{|z|=r} f(z) d \theta= \int _0 ^{2\pi} f(re^{i \theta}) d \theta $ correct?

Also, can I use the ML inequality as follows?

If $|f(z)|<M$ then $|\int _{|z|=r} f(z) d \theta| \leq 2\pi rM$.

Answer 1

The interpretation $$\int_{|z|=r}f(z)\>d\theta=\int_0^{2\pi}f(re^{i\theta})\>d\theta$$ seems correct to me.

If $\bigl|f(z)\bigr|\leq M$ then one obtains $$\left|\int_{|z|=r}f(z)\>d\theta\right|\leq 2\pi\,M\ ,$$ with no $r$ on the RHS, because the total measure of $d\theta$ is $2\pi$, independently of $r$.