# $\int |z| \bar z dz$ over contour

I was wondering whether I can receive some form of confirmation to my method as I feel it's very informal in the way it is worked out.

$$\int_{\Gamma+} |z| \bar z dz$$

Where $\Gamma +$ is a closed contour consisting of the upper semi-circle $|z|=1$, $Im(z)\geq 0$ and the segment $-1\leq Re(z) \leq 1$, $Im(z)=0$ traversed in a positive direction.

The way I worked this out is essentially by splitting the closed contour into 2 contours and evaluating them seperately. My final result is that of $\pi i$. Is my method correct? Or am I able to use some theorems such as Cauchy's Theorem or Green's Theorem? (Both of which I did not use as I am unaware of an easy way to show $f(z)=|z|\bar z$ is either holomorphic, or continuously differentiable)

• Yes, the method is correct. Jan 7, 2017 at 19:02

Since $$|z|=1$$ on the integration path, you also have $$|z|^2=z\overline z=1$$ and thus $$|z|\overline z=(1)(1/z)=1/z$$. So your differential is just $$dz/z$$ and the definite integral is $$\ln [(-1)/(+1)]$$.