Let $f$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$.
Show that $f$ is differentiable on $\mathbb { R } \backslash \{ - 1,1 \}$ and that $$\forall x \in \mathbb { R } \backslash \{ - 1,1 \} \quad f ^ { \prime } ( x ) = \int _ { 0 } ^ { \pi } \frac { 2 x - 2 \cos \theta } { x ^ { 2 } - 2 x \cos \theta + 1 } \mathrm { ~d} \theta$$

Let $f$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$.
Show that $f$ is continuous on $\mathbb { R }$ and that $f ( 1 ) = f ( - 1 ) = 0$.
One may show that $\forall x \in \mathbb { R } , x ^ { 2 } - 2 x \cos \theta + 1 \geqslant \sin ^ { 2 } \theta$ and use the dominated convergence theorem.

For every function $f$ belonging to $\mathcal{F}_{sr}$ and every non-zero natural number $n$, we set $$\left(f * \rho_n\right)(x) = \int_{\mathbb{R}} f(t) \rho_n(x-t) \mathrm{d}t$$
Let $I$ be the function that equals 1 on the interval $[-1,1]$, and 0 elsewhere. For $n \in \mathbb{N}^*$, we set $I_n(x) = I * \rho_n(x)$. a) For $n \in \mathbb{N}^*$ and $x \in \mathbb{R}$, express $I_n(x)$ in terms of $\varphi$. b) For $n \in \mathbb{N}^*$, show that $I_n$ belongs to $\mathcal{D}$ and study its variations. c) Sketch the graphs of $I_2$ and $I_3$. d) Show that the sequence of functions $(I_n)$ converges pointwise to a function $J$ which we shall determine. Show that $J$ and $I$ are equal except on a finite set of points. e) Does the sequence of functions $(I_n)$ converge uniformly to $J$?

If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$ Let $U$ be the function defined by $$\begin{cases} U(x) = 1 & \text{if } x \geqslant 0 \\ U(x) = 0 & \text{if } x < 0 \end{cases}$$ Show that $T_U' = \delta_0$.

If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$
We consider the map $T$ which associates to every function $\varphi$ of $\mathcal{D}$ the real number $T(\varphi)$ defined by $$T(\varphi) = \int_{-1}^{0} t\varphi(t) \mathrm{d}t + \int_{0}^{+\infty} \varphi(t) \mathrm{d}t$$
a) Show that $T$ is a regular distribution. b) Calculate the derivative of this distribution.

If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$ If $f$ is an element of $\mathcal{F}_{sr}$ and if $a$ is a real number, we set $$\lim_{x \rightarrow a^-} f(x) = f(a^-) \quad \text{and} \quad \lim_{x \rightarrow a^+} f(x) = f(a^+)$$ The difference $f(a^+) - f(a^-)$, called the jump at $a$, is denoted $\sigma(a)$. a) Let $a_1, \ldots, a_p$ be real numbers such that $a_1 < \ldots < a_p$. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a piecewise $\mathcal{C}^1$ function. We further assume that $f$ is continuous on $]-\infty, a_1[ \cup ]a_1, a_2[ \cup \ldots \cup ]a_p, +\infty[$. Show that $$T_f' = T_{f'} + \sum_{i=1}^{p} \sigma(a_i) \delta_{a_i}$$ b) Recover by this method the results of questions II.B.3 and II.B.4.b.

We say that the sequence of distributions $(T_n)_{n \in \mathbb{N}}$ converges to the distribution $T$ if $$\forall \varphi \in \mathcal{D}, \lim_{n \rightarrow \infty} T_n(\varphi) = T(\varphi)$$
For $n$ a non-zero natural number, we consider the function $U_n$ zero on the negative reals, affine on the interval $[0, 1/n]$, equal to 1 for reals greater than $1/n$ and continuous on $\mathbb{R}$. a) Show that the sequence of regular distributions $(T_{U_n})_{n \in \mathbb{N}}$ converges to $T_U$. b) Show that $$\forall \varphi \in \mathcal{D} \quad T_{U_n}'(\varphi) = \int_0^{1/n} n\varphi(t) \mathrm{d}t$$ c) Deduce that the distribution $T_{U_n}'$ is regular and give a function $V_n$ such that $T_{V_n} = T_{U_n}'$. d) Sketch $V_n$ for $n = 1, 2, 4$. e) Show that if the sequence of distributions $(T_n)_{n \in \mathbb{N}}$ converges to the distribution $T$, then $(T_n')_{n \in \mathbb{N}}$ converges to $T'$. f) What is the limit of $T_{U_n}'$ as $n$ tends to infinity?

We say that the sequence of distributions $(T_n)_{n \in \mathbb{N}}$ converges to the distribution $T$ if $$\forall \varphi \in \mathcal{D}, \lim_{n \rightarrow \infty} T_n(\varphi) = T(\varphi)$$
For every non-zero natural number $n$, we consider the functions $$\begin{cases} f_n(x) = \dfrac{n}{1 + n^2 x^2} & \\ g_n(x) = nx^n & \text{if } x \in [0,1] \text{ and zero elsewhere} \\ h_n(x) = n^2 \sin nx & \text{if } x \in [-\pi/n, \pi/n] \text{ and zero elsewhere} \end{cases}$$
a) Verify that they belong to $\mathcal{F}_{sr}$. b) Study the variations of the functions $f_n, g_n$ and $h_n$ then sketch their graphs for $n = 1$ and $n = 2$. c) Study the convergence of the sequences of distributions $(T_{f_n}), (T_{g_n})$ and $(T_{h_n})$.

Let $f \in E_{\mathrm{cpm}}$. Show that the function $\mathcal{F}(f)$ is continuous on $\mathbb{R}$.

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; in other words, we need to determine, if any exist, the function or functions $f$ defined and continuous on $\overline{D(0,1)}$ (closed disk), of class $\mathcal{C}^2$ on $D(0,1)$, and such that $$\left\{\begin{array}{l} \Delta f = 0 \text{ on } D(0,1) \\ \forall t \in \mathbb{R},\ f(\cos(t), \sin(t)) = h(t) \end{array}\right.$$ Show the existence and uniqueness of the solution to this Dirichlet problem.

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 the existence and uniqueness of the solution to the Dirichlet problem studied in this part.

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
Show that for all $k \geqslant 0$, if $\varphi \in \mathcal{C}_{c}^{k}(\mathbb{R})$ then $T_{\mu}\varphi \in \mathcal{C}_{c}^{k+1}(\mathbb{R})$. Also show that
$$\left\|(T_{\mu}\varphi)^{(k)}\right\|_{\infty} \leqslant \left\|\varphi^{(k)}\right\|_{\infty}$$

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
We assume that $(\mu_{n})_{n \geqslant 1}$ is a sequence of strictly positive real numbers such that $\sum_{n \geqslant 1} \mu_{n}$ converges. We fix $\psi_{0} \in \mathcal{C}_{c}(\mathbb{R})$ and we define by recursion the sequence $(\psi_{n})_{n \geqslant 0}$ by
$$\forall n \geqslant 0,\quad \psi_{n+1} = T_{\mu_{n+1}} \psi_{n}$$
Show that for all $n \geqslant k$, $\psi_{n}$ is of class $C^{k}$.

Let $f : \mathbb { R } _ { + } \rightarrow \mathbb { R }$, be a piecewise continuous function, strictly positive and integrable. For all $x > 0$, we define
$$g ( x ) = \frac { 1 } { x } \int _ { 0 } ^ { x } t f ( t ) \mathrm { d } t \quad \text { and } \quad h ( x ) = \frac { 1 } { x } g ( x ) = \frac { 1 } { x ^ { 2 } } \int _ { 0 } ^ { x } t f ( t ) \mathrm { d } t .$$
Determine the limit of $g ( x )$ as $x$ tends to 0.

Let $f : \mathbb { R } _ { + } \rightarrow \mathbb { R }$, be a piecewise continuous function, strictly positive and integrable. For all $x > 0$, we define
$$g ( x ) = \frac { 1 } { x } \int _ { 0 } ^ { x } t f ( t ) \mathrm { d } t \quad \text { and } \quad h ( x ) = \frac { 1 } { x } g ( x ) = \frac { 1 } { x ^ { 2 } } \int _ { 0 } ^ { x } t f ( t ) \mathrm { d } t .$$
Determine the limit of $g ( x )$ as $x$ tends to $+ \infty$. Denoting by $\mathbb { 1 } _ { [ 0 , x ] }$ the indicator function of $[ 0 , x ]$, you may note that $g ( x ) = \int _ { 0 } ^ { + \infty } \frac { 1 } { x } t f ( t ) \mathbb { 1 } _ { [ 0 , x ] } ( t ) \mathrm { d } t$.

We fix $( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 }$ and set $\alpha _ { p , q } := \dfrac { p } { q }$. We define, for all $t \in \mathbf { R } _ { + }$, the application $I _ { p , q } : \mathbf { R } _ { + } \rightarrow \mathbf { R }$ by $$I _ { p , q } ( t ) := \int _ { 0 } ^ { 1 } \frac { x ^ { ( t + 1 ) \alpha _ { p , q } } } { 1 + x ^ { \alpha _ { p , q } } } d x$$ Prove that the application $I _ { p , q }$ is well-defined and continuous on $\mathbf { R } _ { + }$.

Let $f : [a,b] \rightarrow [1, \infty)$ be a continuous function and let $g : \mathbb{R} \rightarrow \mathbb{R}$ be defined as $$g(x) = \begin{cases} 0 & \text{if } x < a \\ \int_{a}^{x} f(t)\, dt & \text{if } a \leq x \leq b \\ \int_{a}^{b} f(t)\, dt & \text{if } x > b \end{cases}$$ Then
(A) $g(x)$ is continuous but not differentiable at $a$
(B) $g(x)$ is differentiable on $\mathbb{R}$
(C) $g(x)$ is continuous but not differentiable at $b$
(D) $g(x)$ is continuous and differentiable at either $a$ or $b$ but not both

If $f ( x ) = \left\{ \begin{array} { l l } \int _ { 0 } ^ { x } ( 5 + | 1 - t | ) d t , & x > 2 \\ 5 x + 1 , & x \leq 2 \end{array} \right.$, then
(1) $f ( x )$ is not continuous at $x = 2$
(2) $f ( x )$ is everywhere differentiable
(3) $f ( x )$ is continuous but not differentiable at $x = 2$
(4) $f ( x )$ is not differentiable at $x = 1$