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?

Show an analogous result to Q34 for harmonic functions: for a harmonic function $g$ on $D(0,R)$, show that $\forall r \in [0, R[$, $|g(0)| \leqslant \sup_{t \in \mathbb{R}} |g(r\cos(t), r\sin(t))|$.

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 $D(0,1)$.

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.

Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
Let $T \in M_{n}(\mathbf{R})$. We assume that $M$ is similar to $T$ in $M_{n}(\mathbf{R})$ and we denote by $(\mathrm{S}^{*})$ the differential system $$(\mathrm{S}^{*}) \quad Y' = TY$$
Prove that the coordinates of a solution $X$ of (S) are linear combinations of the coordinates of a solution $Y$ of $(\mathrm{S}^{*})$.

Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
We assume $n = 2$, we then denote $X = (x; y)$ where $x$ and $y$ are two functions differentiable from $\mathbf{R}$ to $\mathbf{R}$ and we set $z = x + iy$.
We assume that there exist real numbers $a$ and $b$ such that $M = \left(\begin{array}{cc} a & b \\ -b & a \end{array}\right)$.
Prove that $X$ is a solution of (S) if and only if $z$ is a solution of a first-order linear differential equation to be determined. Deduce an expression, as a function of $t$, of the coordinates of the solutions of (S).
Solve the system $X' = BX$ where $B$ is the matrix from question 2).

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 $\alpha < 0$.

Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
We assume $n = 2$, we then denote $X = (x; y)$ where $x$ and $y$ are two functions differentiable from $\mathbf{R}$ to $\mathbf{R}$ and we set $z = x + iy$.
Let $M \in M_{2}(\mathbf{R})$ be semi-simple. Give a necessary and sufficient condition, concerning the real and imaginary parts of the eigenvalues of $M$, for every solution of (S) to have each of its coordinates tend to 0 as $+\infty$.

Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
Let $T \in M_{n}(\mathbf{R})$. We assume that $M$ is similar to $T$ in $M_{n}(\mathbf{R})$ and we denote by $(\mathrm{S}^{*})$ the differential system $$(\mathrm{S}^{*}) \quad Y' = TY$$
We consider the following assertions:

• [$\mathbf{A}_{1}$] $\chi_{M}$ is a Hurwitz polynomial;
• [$\mathbf{A}_{2}$] The solutions of (S) tend to $0_{\mathbf{R}^{n}}$ as $t$ tends to $+\infty$;
• [$\mathbf{A}_{3}$] There exist $\alpha > 0$ and $k > 0$ such that for every solution $\Phi$ of (S), $$\forall t \geq 0 \quad : \quad \|\Phi(t)\| \leq k e^{-\alpha t} \|\Phi(0)\|.$$

Let $T \in M_{n}(\mathbf{R})$. We assume that $T$ satisfies the following condition: $$(\mathrm{C}) \quad \exists \beta \in \mathbf{R}_{+}^{*}, \forall X \in \mathbf{R}^{n} : \langle TX, X \rangle \leq -\beta \|X\|^{2}.$$
Prove that $\mathrm{A}_{3}$ is true with $k = 1$ for every solution $\Phi$ of $(\mathrm{S}^{*})$.
Hint: one may introduce the function $t \mapsto e^{2\beta t} \|\Phi(t)\|^{2}$.

Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
We consider the following assertions:

• [$\mathbf{A}_{1}$] $\chi_{M}$ is a Hurwitz polynomial;
• [$\mathbf{A}_{2}$] The solutions of (S) tend to $0_{\mathbf{R}^{n}}$ as $t$ tends to $+\infty$;
• [$\mathbf{A}_{3}$] There exist $\alpha > 0$ and $k > 0$ such that for every solution $\Phi$ of (S), $$\forall t \geq 0 \quad : \quad \|\Phi(t)\| \leq k e^{-\alpha t} \|\Phi(0)\|.$$

Assume that $M \in M_{n}(\mathbf{R})$ is semi-simple. Prove that the assertions $\mathrm{A}_{1}$, $\mathrm{A}_{2}$ and $\mathrm{A}_{3}$ are equivalent.
Hint: one may start with $A_{3}$ implies $A_{2}$.

In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. 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 $e^{tA}$.
$\mathbf{22}$ ▷ Deduce from question 21) that there exists a natural integer $p$ such that, for all $(i,j) \in \llbracket 1,n \rrbracket^2$, we have $$v_{i,j}(t) = O\left(t^p e^{\alpha t}\right) \text{ as } t \rightarrow +\infty.$$

In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. We set $\alpha = \max_{\lambda \in \operatorname{Sp}(A)} \operatorname{Re}(\lambda)$.
$\mathbf{23}$ ▷ Study the converse of question 19): that is, show that if $\alpha < 0$ then $\lim_{t \rightarrow +\infty} f_A(t) = 0_n$.

In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$.
$\mathbf{24}$ ▷ We assume, in this question only, that all eigenvalues of the matrix $A$ have real parts that are positive or zero. Show that, if $X \in \mathbf{C}^n$, we have $$\lim_{t \rightarrow +\infty} e^{tA} X = 0 \Longleftrightarrow X = 0.$$

In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. We introduce the following polynomials: $$\begin{aligned} & P_s(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) < 0}} (X - \lambda)^{m_\lambda}, \\ & P_i(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) > 0}} (X - \lambda)^{m_\lambda}, \\ & P_n(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) = 0}} (X - \lambda)^{m_\lambda}, \end{aligned}$$ and the subspaces $E_s = \operatorname{Ker}(P_s(A))$, $E_i = \operatorname{Ker}(P_i(A))$ and $E_n = \operatorname{Ker}(P_n(A))$ of $E = \mathbf{C}^n$. We have $E = E_s \oplus E_i \oplus E_n$.
$\mathbf{26}$ ▷ Show that $$E_n = \left\{ X \in E \mid \exists C \in \mathbf{R}_+^* \quad \exists p \in \mathbf{N} \quad \forall t \in \mathbf{R} \quad \left\| e^{tA} X \right\|_E \leq C(1 + |t|)^p \right\}.$$

For $p \in \mathbb { R } ^ { * }$ we denote by $(E_p)$ the differential equation on $\mathbb { R } _ { + } ^ { * }$ $$\left( E _ { p } \right) : x \left( y ^ { \prime \prime } - y ^ { \prime } \right) + p y = 0 .$$ Let $p \in \mathbb { R } ^ { * }$ and $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ be a sequence of real numbers. We assume that the power series $\sum _ { n \geqslant 0 } a _ { n } x ^ { n }$ has infinite radius of convergence. Show that the function $f : x \mapsto \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n }$ is a solution of $\left( E _ { p } \right)$ if and only if $$\left\{ \begin{array} { l } a _ { 0 } = 0 \\ n ( n + 1 ) a _ { n + 1 } = ( n - p ) a _ { n } , \quad \forall n \in \mathbb { N } ^ { * } \end{array} \right.$$

For $p \in \mathbb { R } ^ { * }$ we denote by $(E_p)$ the differential equation on $\mathbb { R } _ { + } ^ { * }$ $$\left( E _ { p } \right) : x \left( y ^ { \prime \prime } - y ^ { \prime } \right) + p y = 0 .$$ The coefficients of a power series solution satisfy $a_0 = 0$ and $n(n+1)a_{n+1} = (n-p)a_n$ for all $n \in \mathbb{N}^*$. Show that $(E_p)$ has non-identically zero polynomial solutions if and only if $p \in \mathbb { N } ^ { * }$. Show that then, the non-zero polynomial solutions of $(E_p)$ are of degree $p$ and belong to the vector space $E$.

For $p \in \mathbb { N } ^ { * }$, consider a polynomial $P \in \mathbb { R } [ X ]$ such that the polynomial function $x \mapsto P ( x )$ is a solution of the equation $\left( E _ { p } \right) : x \left( y ^ { \prime \prime } - y ^ { \prime } \right) + p y = 0$. For all $x \in \mathbb { R }$, we denote $h ( x ) = \mathrm { e } ^ { - x } P ( x )$. Show that the function $h$ is a solution of the differential equation $x \left( y ^ { \prime \prime } + y ^ { \prime } \right) + p y = 0$ on $\mathbb { R } _ { + } ^ { * }$.

For $p \in \mathbb { N } ^ { * }$, consider a polynomial $P \in \mathbb { R } [ X ]$ such that the polynomial function $x \mapsto P ( x )$ is a solution of the equation $\left( E _ { p } \right) : x \left( y ^ { \prime \prime } - y ^ { \prime } \right) + p y = 0$. For all $x \in \mathbb { R }$, we denote $h ( x ) = \mathrm { e } ^ { - x } P ( x )$. Justify that the function $h$ is developable as a power series on $\mathbb { R }$.

For $p \in \mathbb { R } ^ { * }$ we denote by $(E_p)$ the differential equation on $\mathbb { R } _ { + } ^ { * }$ $$\left( E _ { p } \right) : x \left( y ^ { \prime \prime } - y ^ { \prime } \right) + p y = 0 .$$ We fix a non-zero real $p$ and assume that $p \notin \mathbb { N } ^ { * }$. The coefficients of a power series solution satisfy $a_0 = 0$ and $n(n+1)a_{n+1} = (n-p)a_n$ for all $n \in \mathbb{N}^*$. Justify the existence of sequences $\left( a _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { R } ^ { \mathbb { N } }$ not identically zero such that the power series $\sum _ { n \geqslant 0 } a _ { n } x ^ { n }$ has infinite radius of convergence and such that the function $x \mapsto \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n }$ is a solution of $(E_p)$.

In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with the projections $p_i$, inclusions $q_i$, $a_i = p_i a q_i$, and $a = \sum _ { i = 1 } ^ { r } q _ { i } a _ { i } p _ { i }$.
Deduce that: $$\forall t \in \mathbf { R } , \quad e ^ { t a } = \sum _ { i = 1 } ^ { r } q _ { i } e ^ { t a _ { i } } p _ { i }$$

In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with $a_i = p_i a q_i$ the endomorphism of $E_i$ and $\|.\|_i$ the norm on $\mathcal{L}(E_i)$.
Show moreover that: $$\forall i \in \llbracket 1 ; r \rrbracket , \quad \forall t \in \mathbf { R } , \quad \left\| e ^ { t a _ { i } } \right\| _ { i } \leqslant \left| e ^ { t \lambda _ { i } } \right| \sum _ { k = 0 } ^ { m _ { i } - 1 } \frac { | t | ^ { k } } { k ! } \left\| a _ { i } - \lambda _ { i } id _ { E _ { i } } \right\| _ { i } ^ { k }$$

In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with $\|.\|_c$ the norm on $\mathcal{L}(\mathbf{C}^n)$.
Deduce the existence of a polynomial $P$ with real coefficients such that: $$\forall t \in \mathbf { R } , \quad \left\| e ^ { t a } \right\| _ { c } \leqslant P ( | t | ) \sum _ { i = 1 } ^ { r } e ^ { t \operatorname { Re } \left( \lambda _ { i } \right) }$$ where $\operatorname { Re } ( z )$ denotes the real part of a complex number $z$.

For any matrix $A \in \mathscr { M } _ { n } ( \mathbf { R } )$, we denote by $u _ { A }$ the endomorphism canonically associated with $A$ in $\mathbf { R } ^ { n }$ and $v _ { A }$ the endomorphism of $\mathbf { C } ^ { n }$ canonically associated with $A$, viewed as a matrix in $\mathscr { M } _ { n } ( \mathbf { C } )$. We keep the notation $\| . \| _ { c }$ for the norm introduced in part A on $\mathcal { L } \left( \mathbf { C } ^ { n } \right)$ and we use $\| . \| _ { r }$ on $\mathcal { L } \left( \mathbf { R } ^ { n } \right)$. Show that there exists $C > 0$ such that: $$\forall A \in \mathscr { M } _ { n } ( \mathbf { R } ) , \quad \forall t \in \mathbf { R } , \quad \left\| e ^ { t u _ { A } } \right\| _ { r } \leqslant C \left\| e ^ { t v _ { A } } \right\| _ { c }$$

We consider $u$ an endomorphism of $\mathbf { R } ^ { n }$, and $A \in \mathscr { M } _ { n } ( \mathbf { R } )$ its matrix in the canonical basis. We denote by $S p ( A )$ the complex spectrum of $A$. Let $g _ { x _ { 0 } }$ be the unique $\mathcal { C } ^ { 1 }$ solution on $\mathbf { R } _ { + }$ of: $$\left\{ \begin{aligned} y ^ { \prime } & = u ( y ) \\ y ( 0 ) & = x _ { 0 } \end{aligned} \right.$$
Show that: $$\forall x _ { 0 } \in \mathbf { R } ^ { n } , \quad \lim _ { t \rightarrow + \infty } \left\| g _ { x _ { 0 } } ( t ) \right\| = 0 \Longleftrightarrow S p ( A ) \subset \mathbf { R } _ { - } ^ { * } + i \mathbf { R } .$$