Integral of $h(z)h'(z)$

Question

If $h$ is analytic on the region below (the two white points are dug out) and $\int_{\alpha}h(z)dz=3$, $\int_{\beta}h(z)dz=4$, then what's $\int_{\gamma}h(z)h'(z)dz$?

I am thinking using $\int_{h(\gamma)}zdz$...but can't find any theorems to apply.

Answer 1

HINT:

Note that $h$ is analytic on $\gamma$ and the antiderivative of $h(z)h'(z)$ is $\frac12 h^2(z)$on $\gamma$.