Let $z$ be a complex variable, and write $x = \Re(z)$ and $y = \Im(z)$ for the real and the imaginary parts, respectively. Let $f(z)$ be a complex polynomial. Let $R > 0$ be a real number and $\gamma$ the circle in $\mathbb{C}$ of radius $R$ and centre at 0, oriented in the counterclockwise direction. What is the value of $$\frac{1}{2\pi \imath R} \int_{\gamma} \left( \Re(f(z))\,\mathrm{d}x + \Im(f(z))\,\mathrm{d}y \right)$$

[7 points] Let $z = e^{\left(\frac{2\pi i}{n}\right)}$. Here $n \geq 2$ is a positive integer, $i^{2} = -1$ and the real number $\frac{2\pi}{n}$ can also be considered as an angle in radians.
(i) Show that $\displaystyle\sum_{k=0}^{n-1} z^{k} = 0$.
(ii) Show that $\displaystyle\sum_{k=0}^{8} \cos(40k+1)^{\circ} = 0$, i.e., $\cos(1^{\circ}) + \cos(41^{\circ}) + \cos(81^{\circ}) + \cos(121^{\circ}) + \cdots + \cos(241^{\circ}) + \cos(281^{\circ}) + \cos(321^{\circ}) = 0$.

Consider the complex polynomial $P ( x ) = x ^ { 6 } + i x ^ { 4 } + 1$. (Here $i$ denotes a square-root of $-1$.) Pick the correct statement(s) from below.
(A) $P$ has at least one real zero.
(B) $P$ has no real zeros.
(C) $P$ has at least three zeros of the form $\alpha + i \beta$ with $\beta < 0$.
(D) $P$ has exactly three zeros $\alpha + i \beta$ with $\beta > 0$.

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 $z \in \mathbb{C}$. We set $z = a + ib$, where $a, b \in \mathbb{R}$.
Let $n \in \mathbb{N}^*$. Determine the modulus and an argument of $\left(1 + \frac{z}{n}\right)^n$ as a function of $a$, $b$ and $n$.

Let $z \in \mathbb{C}$. We set $z = a + ib$, where $a, b \in \mathbb{R}$.
Deduce that $$\lim_{n \rightarrow \infty} \left(1 + \frac{z}{n}\right)^n = e^z$$

Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Justify that $\theta$ and $R$ are well defined.

Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ When $z$ takes successively the values $z_1 = 4$, $z_2 = 2\mathrm{i}$ and $z_3 = 1 - \mathrm{i}\sqrt{3}$, calculate $R(z)$, $\theta(z)$ and $(R(z))^2$.

Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Verify that $\theta(z) \in ]-\pi, \pi[$ and that $R(z) \in \mathcal{P} = \{Z \in \mathbb{C},\, \operatorname{Re}(Z) > 0\}$.

Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Determine $[R(z)]^2$, $\theta \circ R(z)$ and $|z|^{1/2}\mathrm{e}^{\mathrm{i}\theta(z)/2}$ as functions of $z$, $R(z)$ and $\theta(z)$.

Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Solve using $R$ the equation $Z^2 = z$, with unknown $Z \in \mathbb{C}$.

Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Deduce that $R$ is a bijection from $\mathbb{C} \setminus \mathbb{R}^{-}$ to $\mathcal{P}$. Specify its inverse bijection.

Show that $r_1$ and $r_2$ are nonzero and that $r_1/r_2$ belongs to $\mathbb{U}_{n+1}$.

Using the equation (I.1) satisfied by $r_1$ and $r_2$, determine $r_1 r_2$ and $r_1 + r_2$. Deduce that there exists an integer $\ell \in \llbracket 1, n \rrbracket$ and a complex number $\rho$ satisfying $\rho^2 = bc$ such that $$\lambda = a + 2\rho \cos\left(\frac{\ell \pi}{n+1}\right)$$

Deduce that there exists $\alpha \in \mathbb{C}$ such that, for all $k$ in $\llbracket 0, n+1 \rrbracket$, $x_k = 2\mathrm{i}\alpha \frac{\rho^k}{b^k} \sin\left(\frac{\ell k \pi}{n+1}\right)$.

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 $D(0,R)$.

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.

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 $D(0,1)$.