how to deal with $| dz |$ in complex integral, with two examples

Question

I met two integrals:

$$\int_{| z |=1} \frac {|dz |} {z} $$ and

$$\int_{| z |=1} |\frac {dz} {z}| $$.

Actually I have no idea of how to deal with $|dz|$.

Any good suggestions?

Many thanks.

Best

Answer 1

In both cases |dz| is the line element, as ds in differential geometry.

Answer 2

The notation $|dz|$ refers to an integral with respect to arc length.

As far as calculating goes, say $\gamma:[a,b]\to\Bbb C$ is a (sufficiently smooth) curve. Then you know that $$\int_\gamma f(z)dz=\int_a^bf(\gamma(t))\gamma'(t)\,dt.$$Similarly $$\int_\gamma f(z)|dz|=\int_a^bf(\gamma(t))|\gamma'(t)|\,dt.$$