Compute the complex

Question

Compute the complex integral $$\int_{|z|=1} \frac{|dz|}{|2z-1|^2} .$$

I am trying to compute this integral by transferring it to a line integral but I do not really know about the difference between $dz$ and $|dz|$. Could you please show me how to compute this integral? Thank you so much.

Answer 1

$iz|dz| = dz$ and

$$|2z-1|^2 = (2z-1)(2\bar{z}-1) = (2z-1)(2z^{-1}-1)$$

on $|z|=1$. With these combined the integral becomes

$$\int_{|z|=1} \frac{|dz|}{|2z-1|^2} = \frac{1}{i}\int_{|z|=1}\frac{dz}{(2z-1)(2-z)}$$

$$=\frac{1}{2\pi i}\int_{|z|=1} \frac{\frac{\pi}{2-z}}{z-\frac{1}{2}}\:dz = \frac{2\pi}{3}$$

by Cauchy's formula.

Answer 2

For a contour $\gamma:[0,1] \to \mathbb C$, the usual interpretation would be, $$ \int_\gamma f(z) dz = \int_0^1 f(\gamma(t)) ~ \gamma'(t) ~ dt $$ and $$ \int_\gamma f(z) \lvert dz \rvert = \int_0^1 f(z)~\lvert \gamma'(t) \rvert ~ dt $$