Instructor: Dr. Samer Dweik
Instructions:

• Read the questions carefully and make sure you provide all the information that is asked for in the question.

Show all your work. Answers without any explanation or without the correct accompanying work could receive no credit, even if they are correct.

Problem 1.

Ex.1- (a) [3 marks] Fix $a \in \mathbb{C},|a| \geq 1$ and $n \geq 2$. Prove that all zeros of the polynomial $a z^{n}+z+1$ lie inside the disc $D(0,2)$.
(b) [3 marks] Find the number of zeros for the polynomial $z^{4}+8 z^{3}+3 z^{2}+8 z+3$ in the right half-plane.

Problem 2.

Ex.2- (a) [2 marks] Let $f(z)=\sum_{n \geq 0} a_{n} z^{n}$ be a analytic function on the unit disc $D(0,1)$. Assume that $|f(z)| \leq 2020$, for all $z$ such that $|z| \leq 1$. Is it possible that $a_{2021}=2021$ ?
(b) $[2$ marks $]$ Let $f$ be an entire function and $\lim _{z \rightarrow \infty} f(z)=1$. Prove that $f$ is constant.
(c) [2 marks] Let $f$ be an entire function such that $f(z)=f(z+1)=f(z+i)$, for all $z \in \mathbb{C}$. Prove that $f$ is constant.

Problem 3.

Ex.3- Let $D$ be the region outside the circle $(C): x^{2}+(y-2)^{2}=1$ lying above the real axis.
(a) [2 marks] Show that $\alpha=i \sqrt{3}$ and $\alpha^{\star}=-i \sqrt{3}$ are symmetric with respect to the circle
$(C)$ and the real axis.

(b) [ 4 marks] Find a Möbius transformation $f(z)$ that maps $D$ onto an annulus ${a<|z|0$ to be determined.

(c) [3 marks] Solve the Laplace equation $\Delta \varphi=0$ in that region with boundary conditions $\varphi=1$ on the real axis and $\varphi=-1$ on the circle.

Problem 4.

Ex.4-Fix $-1<\alpha<3$. We consider the branch of the logarithm function defined by $\log (z):=$ $\ln |z|+i \arg {-\frac{\pi}{2}}(z)$, where $\left.\left.\arg {-\frac{\pi}{2}}(z) \in\right]-\frac{\pi}{2}, \frac{3 \pi}{2}\right]$ is a branch of $\arg (z)$, and set $z^{\alpha}:=e^{\alpha \log (z)}$
Fix $0<\varepsilon<1^{2}<R<\infty$. Let $\gamma_{R}^{2}$ be the upper half of the circle with centre $z=0$ and radius $R$ and $\gamma_{\varepsilon}$ be the upper half of the circle with centre $z=0$ and radius $\varepsilon .$ Set $s_{\varepsilon, R}^{-}:=[-R,-\varepsilon]$ and $s_{\varepsilon, R}^{+}:=[\varepsilon, R] .$ Let $\Gamma_{\varepsilon, R}$ (traversed counterclockwise) be the contour formed by $\gamma_{R}, \gamma_{\varepsilon}$, $s_{\varepsilon, R}^{+}$ and $s_{\varepsilon, R}^{-} .$ We define
$$I_{\varepsilon, R}:=\int_{\Gamma_{\varepsilon, R}} \frac{z^{\alpha}}{\left(z^{2}+1\right)^{2}} \mathrm{~d} z$$