Practice problems

The following are practice challenge problems for your own use. They will not be collected or marked; the idea is to provide more practice at the sort of problems which might appear as challenge problems on the final exam.

Challenge problem. Let $u$ and $v$ be functions of $r\in [0,\mathrm{\infty })$ and $\theta \in (-\pi ,\pi ]$. Find a system of differential equations which is satisfied by $u$ and $v$ if and only if $f(r{e}^{i\theta })=u(r,\theta )+iv(r,\theta )$, i.e. a polar form of the Cauchy–Riemann equations.

Challenge problem. Let $f$ be an entire function. Evaluate the real-variable integral ${\int }_0^{2\pi }f(e^{i\theta })\cos \theta d\theta$ in terms of $f^{\prime }(0)$.

Challenge problem. Evaluate the integral ${\int }_0^{\mathrm{\infty }}\frac{\sqrt{x}}{x^2+1}𝑑x$ using the residue theorem.

Challenge problem. Let $P(z)$ and $Q(z)$ be polynomials where the degree of $Q$ is at least $2$ greater than the degree of $P$. Prove that the sum of all the residues of the rational function $f(z)=\frac{P(z)}{Q(z)}$ in the entire complex plane is exactly $0$.

Challenge problem. Prove that for any real number $\lambda >1$, the equation $z=\lambda -e^{-z}$ has exactly one solution in the right half-plane $\mathrm{Re}(z)>0$.

Challenge problem. Let . Show that there is no analytic bijection $f:D\to ℂ$.