For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$, and $F$ satisfies $F^{\prime} + AF = 0$ on $\mathbb{R}$. Deduce an expression for $F(x)$.
You may start by differentiating the function $x \mapsto -\frac{1}{8} \ln\left(1 + x^{2}\right) + \frac{\mathrm{i}}{4} \arctan x$.

Let $\lambda \in \mathbb{C}$ such that $\operatorname{Re}(\lambda) > 0$. Let $u$ be a function with complex values of class $\mathcal{C}^{1}$ on $\mathbb{R}^{+}$.
Suppose that the function $v = u' + \lambda u$ is bounded on $\mathbb{R}^{+}$. Show that $u$ is bounded on $\mathbb{R}^{+}$.
One may consider the differential equation $y' + \lambda y = v$.

Let $T \in \mathcal{M}_{n}(\mathbb{C})$ be an upper triangular matrix with complex entries. Suppose that the diagonal entries of $T$ are complex numbers with strictly positive real part. Let $u_{1}, \ldots, u_{n}$ be functions with complex values, defined and of class $\mathcal{C}^{1}$ on $\mathbb{R}^{+}$ and let, for all $t \in \mathbb{R}^{+}$, $$U(t) = \begin{pmatrix} u_{1}(t) \\ \vdots \\ u_{n}(t) \end{pmatrix}$$
Suppose that, for all $t \in \mathbb{R}^{+}$, $U'(t) + TU(t) = 0$.
Show that the functions $u_{j}$, where $1 \leqslant j \leqslant n$, are bounded on $\mathbb{R}^{+}$.

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part. Recall that for any matrix $M \in \mathcal{M}_{n}(\mathbb{C})$, $\exp(M) = \sum_{k=0}^{\infty} \frac{M^{k}}{k!}$.
Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix with complex eigenvalues $\lambda_{1}, \ldots, \lambda_{n}$ and let $\alpha$ be a real number such that $0 < \alpha < \min_{1 \leqslant j \leqslant n} \operatorname{Re}(\lambda_{j})$.
Show that the function $t \mapsto \mathrm{e}^{\alpha t}\exp(-tA)$ is bounded on $\mathbb{R}^{+}$.
One may apply question III.B.2 to an upper triangular matrix $T$ similar to $A - \alpha I_{n}$.

Let $\lambda \in \mathbb{C}$ such that $\operatorname{Re}(\lambda) > 0$. Let $u$ be a function with complex values of class $\mathcal{C}^{1}$ on $\mathbb{R}^{+}$.
Suppose that the function $v = u^{\prime} + \lambda u$ is bounded on $\mathbb{R}^{+}$. Show that $u$ is bounded on $\mathbb{R}^{+}$.
One may consider the differential equation $y^{\prime} + \lambda y = v$.

Let $T \in \mathcal{M}_{n}(\mathbb{C})$ be an upper triangular matrix with complex entries. Suppose that the diagonal entries of $T$ are complex numbers with strictly positive real part. Let $u_{1}, \ldots, u_{n}$ be functions with complex values, defined and of class $\mathcal{C}^{1}$ on $\mathbb{R}^{+}$ and let, for all $t \in \mathbb{R}^{+}$, $$U(t) = \left(\begin{array}{c} u_{1}(t) \\ \vdots \\ u_{n}(t) \end{array}\right)$$ Suppose that, for all $t \in \mathbb{R}^{+}, U^{\prime}(t) + T U(t) = 0$.
Show that the functions $u_{j}$, where $1 \leqslant j \leqslant n$, are bounded on $\mathbb{R}^{+}$.

Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix with complex eigenvalues $\lambda_{1}, \ldots, \lambda_{n}$ and let $\alpha$ be a real number such that $0 < \alpha < \min_{1 \leqslant j \leqslant n} \operatorname{Re}\left(\lambda_{j}\right)$.
Show that the function $t \mapsto \mathrm{e}^{\alpha t} \exp(-tA)$ is bounded on $\mathbb{R}^{+}$.
One may apply question III.B.2 to an upper triangular matrix $T$ similar to $A - \alpha I_{n}$.

Let $\lambda \in \mathbb { R }$. Show that the problem
$$\left\{ \begin{array} { l } - v _ { \lambda } ^ { \prime \prime } ( x ) + c ( x ) v _ { \lambda } ( x ) = f ( x ) , x \in [ 0,1 ] \\ v _ { \lambda } ( 0 ) = 0 \\ v _ { \lambda } ^ { \prime } ( 0 ) = \lambda \end{array} \right.$$
admits a unique solution $v _ { \lambda } \in \mathcal { C } ^ { 2 } ( [ 0,1 ] , \mathbb { R } )$.

Show that for all $\lambda \in \mathbb { R } , v _ { \lambda }$ can be expressed in the form:
$$v _ { \lambda } = \lambda w _ { 1 } + w _ { 2 }$$
with $w _ { 1 } \in \mathcal { C } ^ { 2 } ( [ 0,1 ] , \mathbb { R } )$ the unique solution of the system
$$\left\{ \begin{array} { l } - w _ { 1 } ^ { \prime \prime } ( x ) + c ( x ) w _ { 1 } ( x ) = 0 , x \in [ 0,1 ] \\ w _ { 1 } ( 0 ) = 0 \\ w _ { 1 } ^ { \prime } ( 0 ) = 1 \end{array} \right.$$
and $w _ { 2 }$ a function independent of $\lambda$ to be characterized.

Show that $w _ { 1 } ( 1 ) \neq 0$.

Deduce that there exists a solution $u \in \mathcal { C } ^ { 2 } ( [ 0,1 ] , \mathbb { R } )$ to problem (1): $$\left\{ \begin{array} { l } - u ^ { \prime \prime } ( x ) + c ( x ) u ( x ) = f ( x ) , x \in [ 0,1 ] \\ u ( 0 ) = u ( 1 ) = 0 \end{array} \right.$$ Show that this solution is unique.

A tridiagonal matrix is a Toeplitz matrix of the form $T(0,\ldots,0,t_{-1},t_0,t_1,0,\ldots,0)$, i.e. a matrix of the form $$A_n(a,b,c) = \left(\begin{array}{cccc} a & b & & (0) \\ c & a & \ddots & \\ & \ddots & \ddots & b \\ (0) & & c & a \end{array}\right)$$ where $(a,b,c)$ are complex numbers. We fix $(a,b,c)$ three complex numbers such that $bc \neq 0$. Let $\lambda \in \mathbb{C}$ be an eigenvalue of $A_n(a,b,c)$ and $X = \left(\begin{array}{c} x_1 \\ \vdots \\ x_n \end{array}\right) \in \mathbb{C}^n$ be an associated eigenvector.
Show that if we set $x_0 = 0$ and $x_{n+1} = 0$, then $(x_1, \ldots, x_n)$ are the terms of rank varying from 1 to $n$ of a sequence $(x_k)_{k \in \mathbb{N}}$ satisfying $x_0 = 0, x_{n+1} = 0$ and $$\forall k \in \mathbb{N}, \quad bx_{k+2} + (a-\lambda)x_{k+1} + cx_k = 0$$

Recall the expression of the general term of the sequence $(x_k)_{k \in \mathbb{N}}$ as a function of the solutions of the equation $$bx^2 + (a-\lambda)x + c = 0 \tag{I.1}$$

Using the conditions imposed on $x_0$ and $x_{n+1}$, show that (I.1) admits two distinct solutions $r_1$ and $r_2$.

We now assume $\lambda \neq 0$. Give a necessary and sufficient condition for the differential equation (II.2) $$z''(\theta) + \lambda z(\theta) = 0$$ to admit non-zero $2\pi$-periodic solutions. Give these solutions.

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We now assume $\lambda \neq 0$. Give a necessary and sufficient condition for (II.2) $$z''(\theta) + \lambda z(\theta) = 0$$ to admit non-zero $2\pi$-periodic solutions. Give these solutions.

We now assume $\lambda \neq 0$. Solve the differential equation (II.1) $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$ on $\mathbb{R}^{+*}$. One may consider, justifying its existence, a function $Z$ of class $\mathcal{C}^2$ on $\mathbb{R}$ such that, for all $r > 0$, $z(r) = Z(\ln(r))$.

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We now assume $\lambda \neq 0$. Solve (II.1) on $\mathbb{R}^{+*}$: $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$ One may consider, justifying its existence, a function $Z$ of class $\mathcal{C}^2$ on $\mathbb{R}$ such that, for all $r > 0$, $z(r) = Z(\ln(r))$.

We now assume $\lambda \neq 0$. Among the solutions of the differential equation (II.1) $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$ which are the solutions that extend continuously to 0?

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We now assume $\lambda \neq 0$. Which are the solutions of (II.1) that extend continuously to 0?

For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t)\,\mathrm{d}t$$ Let $\lambda \in \mathbb{R}$ be a nonzero eigenvalue of $T$ and $f$ be an associated eigenvector. Show that $f$ is a solution of the differential equation $\lambda f'' = -f$.

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t) \, \mathrm{d}t$$ Let $\lambda \in \mathbb{R}$ be a nonzero eigenvalue of $T$ and $f$ be an associated eigenvector. Show that $f$ is a solution of the differential equation $\lambda f'' = -f$.

Let two real numbers $a$ and $c$ such that $c \in D$. Determine the solutions expandable as power series of the differential equation $$x y''(x) + (c - x) y'(x) - a y(x) = 0.$$ We will express these solutions using the Pochhammer symbol and specify the algebraic structure of their set.

To each function $f \in E$, we associate the function $U ( f )$ with derivative $U(f)^\prime(x) = \mathrm { e } ^ { x } \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Let $f \in E$. Show that $U ( f )$ is of class $\mathcal { C } ^ { 2 }$ on $\mathbb { R } _ { + } ^ { * }$ and that the function $U ( f )$ is a solution on $\mathbb { R } _ { + } ^ { * }$ of the differential equation $$y ^ { \prime \prime } - y ^ { \prime } = - \frac { f ( x ) } { x }$$

In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$ a square matrix with complex coefficients, and we denote by $u$ the endomorphism of $\mathbf{C}^n$ canonically associated with this matrix. We set $\alpha = \max_{\lambda \in \operatorname{Sp}(A)} \operatorname{Re}(\lambda)$. For all real $t$ and for $(i,j) \in \llbracket 1,n \rrbracket^2$, we denote by $v_{i,j}(t)$ the coefficient with indices $(i,j)$ of the matrix $e^{tA}$.
$\mathbf{19}$ ▷ Show that, if $\lim_{t \rightarrow +\infty} f_A(t) = 0_n$, then $\alpha < 0$.