Find an analytic function $F$ with
$F'(z)=f(z)$ where $f=\cosh (z)$.
Show that if $f$ is analytic on a domain $D$ and $g$ is analytic on a domain $G$, containing the range of $f$, then their composition $g(f(z))$ is analytic on D, and the chain rule holds:
$(g(f(z))' = g'(f(z))f'(z)$.
Let $\gamma$ be a piecewise smooth simple closed curve, and suppose that $F$ is analytic on some domain containing $\gamma$. Show that
$$\int_\gamma F'(z)\,dz=0.$$
Show that the Cauchy-Riemann equations in polar coordinates are
$$
\left\{\begin{aligned}
&r\frac{\partial u}{\partial r}=\frac{\partial v}{\partial \theta},\\
&r\frac{\partial v}{\partial r}=-\frac{\partial u}{\partial \theta}.
\end{aligned}\right.$$