Let $f(z)$ be a power-series (with complex coefficients) centred at $0 \in \mathbb{C}$ and with a radius of convergence 2. Suppose that $f(0) = 0$. Choose the correct statement(s) from below:
(A) $f^{-1}(0) = \{0\}$;
(B) If $f$ is a non-constant function on $\{|z| < 2\}$, then $f^{-1}(0) = \{0\}$;
(C) If $f$ is a non-constant function, then for all $\zeta \in \mathbb{C}$ with sufficiently small $|\zeta|$, the equation $f(z) = \zeta$ has a solution;
(D) $$\int_{\gamma} f^{(n)}(z)\,\mathrm{d}z = 0$$ for every $n \geq 1$, where $\gamma$ is a unit circle centred at 0, oriented clockwise, and $f^{(n)}$ is the $n$th derivative of $f(z)$.

Let $f ( z ) = \sum _ { n \geq 0 } a _ { n } z ^ { n }$ be an analytic function on the open unit disc $D$ around 0 with $a _ { 1 } \neq 0$. Suppose that $\sum _ { n \geq 2 } \left| n a _ { n } \right| < \left| a _ { 1 } \right|$. Then which of the following are true?
(A) There are only finitely many such $f$.
(B) $\left| f ^ { \prime } ( z ) \right| > 0$ for all $z \in D$.
(C) If $z , w \in D$ are such that $z \neq w$ and $f ( z ) = f ( w )$, then $a _ { 1 } = - \sum _ { n \geq 2 } a _ { n } \left( z ^ { n - 1 } + z ^ { n - 2 } w + \cdots + w ^ { n - 1 } \right)$.
(D) $f$ is one-one on $D$.

Let $U$ and $V$ be non-empty open connected subsets of $\mathbb { C }$ and $f : U \longrightarrow V$ an analytic function. Suppose that for all compact subsets $K$ of $V , f ^ { - 1 } ( K )$ is compact. Show that $f ( U ) = V$.

Let $f$ be an entire function such that $f$ maps the open unit ball $D$ around 0 to itself. Suppose further that $f ( 0 ) = 0$ and $f ( 1 ) = 1$. Show that $f ^ { \prime } ( 1 ) \in \mathbb { R }$ and that $\left| f ^ { \prime } ( 1 ) \right| \geq 1$.

Pick the correct statement(s) from below.
(A) If $f ( z )$ is a function defined on $\mathbb { C }$ that satisfies the Cauchy-Riemann equations at $z = 0$, then $f ( z )$ is complex-differentiable at $z = 0$.
(B) The function $\frac { ( \sin z - z ) \bar { z } ^ { 3 } } { | z | ^ { 6 } }$ is holomorphic on $\{ z \in \mathbb{C} : 0 < | z | < 1 \}$ and has a removable singularity at $z = 0$.
(C) There exists a holomorphic function on $\{ z \in \mathbb { C } : | z | > 3 \}$ whose derivative is $\frac { z } { ( z - 2 ) ^ { 2 } }$.
(D) There exists a holomorphic function on the upper half plane $\{ z \in \mathbb { C } : \mathfrak { I } z > 0 \}$ whose derivative is $\frac { z } { ( z - 2 ) ^ { 2 } \left( z ^ { 2 } + 4 \right) }$.

Prove or disprove the following statements:
(A) (5 marks) Suppose that $f ( z )$ is a complex analytic function in the punctured unit disk $0 < | z | < 1$ such that $\lim _ { n \longrightarrow \infty } f \left( \frac { 1 } { n } \right) = 0$ and $\lim _ { n \longrightarrow \infty } f \left( \frac { 2 } { 2 n - 1 } \right) = 1$, then there exists a positive integer $N > 0$ such that $\lim _ { z \rightarrow 0 } \left| z ^ { - N } f ( z ) \right| = \infty$.
(B) (5 marks) There exists a non-zero entire function $f$ such that $f \left( e ^ { 2 \pi i e n ! } \right) = 0$ for all $n \geq 2025$.

Show that the power series $\sum _ { n = 1 } ^ { \infty } z ^ { n ! }$ represents an analytic function $f ( z )$ in the open unit disk $\Delta$ centred at 0. Show that $f ( z )$ cannot be extended to a continuous function on any connected open set $U$ such that $U$ is strictly larger than $\Delta$.

Let $f$ be a function that expands as a power series on $D(0,R)$. Show that if $|f|$ attains a maximum at 0, then $f$ is constant on $D(0,R)$.

Show the d'Alembert-Gauss theorem: every non-constant complex polynomial has at least one root.
One may proceed by contradiction, assume that there exists a polynomial that does not vanish, and consider its inverse.

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, set $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z)\, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that the function $z \mapsto \frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}$ is expandable as a power series for $|z| < 1$ and calculate its power series expansion. Deduce that the function $(x,y) \mapsto g(x + \mathrm{i}y)$ is a harmonic function on $D(0,1)$.

Throughout this part, $f$ denotes a function that expands in a power series on $D(0,R)$, i.e., $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ Show that if $|f|$ attains a maximum at 0, then $f$ is constant on $D(0,R)$.

Show the d'Alembert-Gauss theorem: every non-constant complex polynomial has at least one root.
One may proceed by contradiction, assume that there exists a polynomial that does not vanish, and consider its inverse.

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. We seek to solve the Dirichlet problem on the unit disk; we need to determine the function or functions $f$ defined and continuous on $\overline{D(0,1)}$, of class $\mathcal{C}^2$ on $D(0,1)$, and such that $$\begin{cases} \Delta f = 0 \text{ on } D(0,1) \\ \forall t \in \mathbb{R}, f(\cos(t), \sin(t)) = h(t) \end{cases}$$ For this, we set, for any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that the function $z \mapsto \frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}$ is expandable in a power series for $|z| < 1$ and calculate its power series expansion. Deduce that the function $(x,y) \mapsto g(x + \mathrm{i}y)$ is a harmonic function on $D(0,1)$.

We are given a polynomial $P \in \mathbb{C}[X]$ of degree $d$. Deduce from question 3.17 that: $$\|P\|_{\mathbb{D}} = \|P\|_{\partial\mathbb{D}}$$ One may apply question 3.17 to an element $z \in \mathbb{D}$ such that $|P(z)| = \|P\|_{\mathbb{D}}$.

Let $f ( t )$ be a periodic function of period $T , f ( t + T ) = f ( t ) ( T > 0 )$, and be expanded in the complex Fourier series as follows:
$$f ( t ) = \sum _ { n = - \infty } ^ { \infty } F _ { n } \exp \left( - i \omega _ { n } t \right)$$
Here, $i$ is the imaginary unit, and $t$ is a real number. Answer the following questions.
I. Express $\omega _ { n }$ using $T$ and $n$.
II. Let $M$ be a positive-integer constant and $\delta ( t )$ be the delta function, and define
$$\hat { f } ( t ) = \sum _ { m = 0 } ^ { M - 1 } f ( t ) \delta ( t - m \Delta t ) , \quad \Delta t = \frac { T } { M }$$
Express
$$\lim _ { \varepsilon \rightarrow + 0 } \int _ { - \varepsilon } ^ { T - \varepsilon } \hat { f } ( t ) \exp \left( i \omega _ { k } t \right) d t$$
using $F _ { n } ( n = - \infty , \cdots , - 1,0,1 , \cdots , \infty )$. Here $k$ is an arbitrary integer.
III. $\Delta t$ is given in Question II. Express $F _ { j } ( j = 0,1,2 , \cdots , M - 1 )$ using
$$f ( 0 ) , \quad f ( \Delta t ) , \quad f ( 2 \Delta t ) , \cdots , \quad f ( ( M - 1 ) \Delta t )$$
when $F _ { n } = 0 ( n < 0$ or $n \geq M )$.
IV. Calculate $F _ { j }$ in Question III when $f ( l \Delta t ) = ( - 1 ) ^ { l } \quad ( l = 0,1,2 , \cdots , M - 1 )$.