Exercise 4 — Candidates who have not followed the specialization course
We define the sequence of complex numbers $( z _ { n } )$ in the following way: $z _ { 0 } = 1$ and, for every natural integer $n$,
$$z _ { n + 1 } = \frac { 1 } { 3 } z _ { n } + \frac { 2 } { 3 } \mathrm { i } .$$
We place ourselves in a plane with an orthonormal direct coordinate system $( \mathrm { O } ; \vec { u } , \vec { v } )$. For every natural integer $n$, we denote $\mathrm { A } _ { n }$ the point in the plane with affix $z _ { n }$. For every natural integer $n$, we set $u _ { n } = z _ { n } - \mathrm { i }$ and we denote $\mathrm { B } _ { n }$ the point with affix $u _ { n }$. We denote C the point with affix i.

1. Express $u _ { n + 1 }$ as a function of $u _ { n }$, for every natural integer $n$.
2. Prove that, for every natural integer $n$,

$$u _ { n } = \left( \frac { 1 } { 3 } \right) ^ { n } ( 1 - \mathrm { i } ) .$$

1. a. For every natural integer $n$, calculate, as a function of $n$, the modulus of $u _ { n }$. b. Prove that

$$\lim _ { n \rightarrow + \infty } \left| z _ { n } - \mathrm { i } \right| = 0$$
c. What geometric interpretation can be given of this result?
4. a. Let $n$ be a natural integer. Determine an argument of $u _ { n }$. b. Prove that, as $n$ ranges over the set of natural integers, the points $\mathrm { B } _ { n }$ are collinear. c. Prove that, for every natural integer $n$, the point $\mathrm { A } _ { n }$ belongs to the line with reduced equation:
$$y = - x + 1 .$$

Consider the equation $( E ) : z ^ { 3 } = 4 z ^ { 2 } - 8 z + 8$ with unknown complex number $z$.
a. Prove that, for all complex numbers $z$, $$z ^ { 3 } - 4 z ^ { 2 } + 8 z - 8 = ( z - 2 ) \left( z ^ { 2 } - 2 z + 4 \right) .$$
b. Solve equation ( $E$ ).
c. Write the solutions of equation ( $E$ ) in exponential form.
The complex plane is equipped with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ). Let $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D be the four points with respective affixes $$z _ { \mathrm { A } } = 1 + \mathrm { i } \sqrt { 3 } \quad z _ { \mathrm { B } } = 2 \quad z _ { \mathrm { C } } = 1 - \mathrm { i } \sqrt { 3 } \quad z _ { \mathrm { D } } = 1 .$$
2. What is the nature of quadrilateral OABC? Justify.
3. Let M be the point with affix $z _ { \mathrm { M } } = \frac { 7 } { 4 } + \mathrm { i } \frac { \sqrt { 3 } } { 4 }$.
a. Prove that points $\mathrm { A } , \mathrm { M }$ and B are collinear.
b. Prove that triangle DMB is right-angled.

Consider the equation $$z^{2018} = 2018^{2018} + i$$ where $i = \sqrt{-1}$.
(a) How many complex solutions does this equation have?
(b) How many solutions lie in the first quadrant?
(c) How many solutions lie in the second quadrant?

Let $f ( z ) = z ^ { 7 } - 4 z ^ { 3 } - 11$. Pick the correct statement(s) from below.
(A) $f ( z )$ has at least 1 zero in the open set $\{ | z | > 2 \}$.
(B) $f ( z )$ has at least 5 zeroes in the annular region $\{ 1 < | z | < 2 \}$.
(C) $f ( z )$ has exactly 6 zeroes in the annular region $\{ 1 < | z | < 2 \}$.
(D) $f ( z )$ has exactly 1 zero in the closed disc $\{ | z | \leq 1 \}$.

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 $x \in [ 0,1 [$ and $\theta \in \mathbf { R }$. Using the function $L$, show that
$$\left| \frac { 1 - x } { 1 - x e ^ { i \theta } } \right| \leq \exp ( - ( 1 - \cos \theta ) x )$$
Deduce that for all $x \in [ 0,1 [$ and all real $\theta$,
$$\left| \frac { P \left( x e ^ { i \theta } \right) } { P ( x ) } \right| \leq \exp \left( - \frac { 1 } { 1 - x } + \operatorname { Re } \left( \frac { 1 } { 1 - x e ^ { i \theta } } \right) \right)$$

Let $x \in \left[ \frac { 1 } { 2 } , 1 [ \right.$ and $\theta \in \mathbf { R }$. Show that
$$\frac { 1 } { 1 - x } - \operatorname { Re } \left( \frac { 1 } { 1 - x e ^ { i \theta } } \right) \geq \frac { x ( 1 - \cos \theta ) } { ( 1 - x ) \left( ( 1 - x ) ^ { 2 } + 2 x ( 1 - \cos \theta ) \right) }$$
Deduce that
$$\left| \frac { P \left( x e ^ { i \theta } \right) } { P ( x ) } \right| \leq \exp \left( - \frac { 1 - \cos \theta } { 6 ( 1 - x ) ^ { 3 } } \right) \quad \text { or } \quad \left| \frac { P \left( x e ^ { i \theta } \right) } { P ( x ) } \right| \leq \exp \left( - \frac { 1 } { 3 ( 1 - x ) } \right)$$

We denote $\omega(t) = e^{2i\pi t}$ for $t \in [0,1]$. For all $p \in \mathbb{Z}$, we set $$I_{p} = \int_{0}^{1} \frac{\omega(t)^{p+1}}{\mathrm{e}^{\omega(t)} - 1} \,\mathrm{d}t.$$ Verify that this integral is well defined for all $p \in \mathbb{Z}$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ For $a \in \mathbb{R}$, we define the application $$\begin{array}{ccc} Z_a : [0,1] & \longrightarrow & \mathbb{C} \\ t & \longmapsto & t + iat(1-t). \end{array}$$ Show that the application $$\begin{array}{clc} ]0,1[ \times \mathbb{R} & \longrightarrow & \mathbb{C} \\ (t,a) & \longmapsto & Z_a(t) \end{array}$$ is injective.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ For $a \in \mathbb{R}$, we define the application $$\begin{array}{ccc} Z_a : [0,1] & \longrightarrow & \mathbb{C} \\ t & \longmapsto & t + iat(1-t). \end{array}$$ Let $M_1$ and $M_2$ be two elements of $(\mathbb{C}[A])^*$. Show that there exists $a \in \mathbb{R}$ such that $$\forall t \in [0,1], \quad M(t) = Z_a(t) M_1 + \left(1 - Z_a(t)\right) M_2 \in (\mathbb{C}[A])^*.$$

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ For $a \in \mathbb{R}$, we define the application $$\begin{array}{ccc} Z_a : [0,1] & \longrightarrow & \mathbb{C} \\ t & \longmapsto & t + iat(1-t). \end{array}$$ Using the result of question 10b, deduce that $(\mathbb{C}[A])^*$ is path-connected.

We denote $\mathbb{U}$ the multiplicative group of complex numbers of modulus 1. Show that there exist a function $\beta \in \mathcal{E}$ and a constant $C \in ]0,1[$ such that, for all $\zeta \in \mathbb{U}$, $$\mathrm{e}^{\zeta} - 1 = \zeta(1 + \zeta \beta(\zeta)) \quad \text{and} \quad |\beta(\zeta)| \leqslant C.$$

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Show that there exists an open set $U$ of $\mathbb{C}[A]$ containing $0$ and an open set $V$ of $\mathbb{C}[A]$ containing the identity matrix $I_n$ such that the exponential function induces a continuous bijection from $U \subset \mathbb{C}[A]$ to $V$ whose inverse is a continuous function on $V$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Using the result of question 11a, deduce that $\exp(\mathbb{C}[A])$ is an open set of $\mathbb{C}[A]$.

Using the result of Q11, deduce that for all $\zeta \in \mathbb{U}$ and all $p \in \mathbb{Z}$, $$\frac{\zeta^{p}}{\mathrm{e}^{\zeta} - 1} = \sum_{j=0}^{+\infty} (-1)^{j} \zeta^{j+p-1} \beta(\zeta)^{j}$$ where $\beta \in \mathcal{E}$ and $|\beta(\zeta)| \leqslant C < 1$ for all $\zeta \in \mathbb{U}$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Show that $\exp(\mathbb{C}[A])$ is a closed set of $(\mathbb{C}[A])^*$.

We denote $\omega(t) = e^{2i\pi t}$ for $t \in [0,1]$, and for all $p \in \mathbb{Z}$, $$I_{p} = \int_{0}^{1} \frac{\omega(t)^{p+1}}{\mathrm{e}^{\omega(t)} - 1} \,\mathrm{d}t.$$ Show that $I_{0} = 1$ and that, for all $p \in \mathbb{N}^{*}$, $I_{p} = 0$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ We want to show that $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$. We suppose that $\exp(\mathbb{C}[A]) \neq (\mathbb{C}[A])^*$ and we fix $M_1, M_2 \in (\mathbb{C}[A])^*$ such that $M_1 \in \exp(\mathbb{C}[A])$ and $M_2 \notin \exp(\mathbb{C}[A])$.
Show that there exists a continuous map $f$ from $(\mathbb{C}[A])^*$ to $\{0,1\}$ such that $f(M_1) = 0$ and $f(M_2) = 1$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ We want to show that $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$. We suppose that $\exp(\mathbb{C}[A]) \neq (\mathbb{C}[A])^*$ and we fix $M_1, M_2 \in (\mathbb{C}[A])^*$ such that $M_1 \in \exp(\mathbb{C}[A])$ and $M_2 \notin \exp(\mathbb{C}[A])$.
Using the result of question 13a, conclude that $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$.

For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, we define $$B_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\omega(t)}}{(\mathrm{e}^{\omega(t)} - 1)\omega(t)^{n-1}} \,\mathrm{d}t$$ where $\omega(t) = e^{2i\pi t}$, and for all $p \in \mathbb{Z}$, $$I_{p} = \int_{0}^{1} \frac{\omega(t)^{p+1}}{\mathrm{e}^{\omega(t)} - 1} \,\mathrm{d}t.$$ Show that, for all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, $$B_{n}(z) = n! \sum_{k=0}^{n} \frac{z^{k}}{k!} I_{k-n}.$$ Deduce that $B_{n}$ is a monic polynomial of degree $n$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ Using the result $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$, conclude that $\exp(\mathscr{M}_n(\mathbb{C})) = \mathrm{GL}_n(\mathbb{C})$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. Using the result $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$, conclude that $\exp(\mathscr{M}_n(\mathbb{C})) = \mathrm{GL}_n(\mathbb{C})$.

For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, the Bernoulli polynomial is defined by $$B_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\omega(t)}}{(\mathrm{e}^{\omega(t)} - 1)\omega(t)^{n-1}} \,\mathrm{d}t$$ where $\omega(t) = e^{2i\pi t}$. Show that, for all $n \in \mathbb{N}^{*}$, $B_{n}' = n B_{n-1}$.

For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, the Bernoulli polynomial is defined by $$B_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\omega(t)}}{(\mathrm{e}^{\omega(t)} - 1)\omega(t)^{n-1}} \,\mathrm{d}t$$ where $\omega(t) = e^{2i\pi t}$. Show that, for all $n \in \mathbb{N}^{*}$ and all $z \in \mathbb{C}$, $$B_{n}(z+1) - B_{n}(z) = n z^{n-1}.$$

Using the Bernoulli polynomials $(B_n)_{n \in \mathbb{N}}$ satisfying $B_n(z+1) - B_n(z) = nz^{n-1}$ for all $n \in \mathbb{N}^*$, deduce the expression of a polynomial function satisfying the equation $(E_h)$: $$\forall x \in \mathbb{C},\, f(x+1) - f(x) = h(x)$$ on $\mathbb{C}$ when $h$ is a polynomial function.