For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$, where $\mathrm{i}$ denotes the complex number with modulus 1 and argument $\pi/2$.
Show that the function $F : \begin{aligned} & \mathbb{R} \rightarrow \mathbb{C} \\ & x \mapsto F(x) \end{aligned}$ is defined and of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$.
Let $k$ be a non-zero natural number and let $x$ be a real number. Give an integral expression for $F^{(k)}(x)$, the $k$-th derivative of $F$ at $x$. Specify $F(0)$.

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$. Prove that $F$ satisfies on $\mathbb{R}$ a differential equation of the form $F^{\prime} + AF = 0$, where $A$ is a function to be specified.

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 $x \mapsto -\frac{1}{8} \ln\left(1 + x^{2}\right) + \frac{\mathrm{i}}{4} \arctan x$.

We consider the function $\theta : \mathbb{R} \rightarrow \mathbb{C}$ defined by $\theta(x) = \exp(-\pi x^{2})$, for $x \in \mathbb{R}$.
I.E.1) Justify that $\theta$ belongs to $\mathcal{S}$ and that $\mathcal{F}(\theta)$ is a solution of the differential equation
$$\forall \xi \in \mathbb{R}, \quad y'(\xi) = -2\pi\xi\, y(\xi)$$
I.E.2) Establish that $\mathcal{F}(\theta) = \theta$.
We will admit that $\int_{-\infty}^{+\infty} \theta(x) \mathrm{d}x = 1$.

We denote $L$ the operator that associates to a function $f : \mathbb { R } \rightarrow \mathbb { R }$ of class $\mathscr { C } ^ { 2 }$, the function $Lf$ defined by $$\forall x \in \mathbb { R } , \quad L f ( x ) = \frac { 1 } { 2 } f ^ { \prime \prime } ( x ) - x f ^ { \prime } ( x )$$ We recall that the measure $\mu$ is defined by $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$.
3a. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be of class $\mathscr { C } ^ { 2 }$. Show that $L f = \frac { 1 } { 2 \mu } \left( \mu f ^ { \prime } \right) ^ { \prime }$.
3b. Let $h _ { 1 } , h _ { 2 }$ be two functions in $\mathscr { C } _ { b } ^ { 2 }$. Show that $$\int h _ { 1 } ( x ) \left( L h _ { 2 } \right) ( x ) \mu ( x ) d x = - \frac { 1 } { 2 } \int h _ { 1 } ^ { \prime } ( x ) h _ { 2 } ^ { \prime } ( x ) \mu ( x ) d x$$ after having justified the existence of each term of the formula.

We consider a function $f \in \mathscr { C } _ { b } ^ { 0 }$. We define for $( t , x ) \in \mathbb { R } ^ { 2 }$ $$\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$$ where $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$. Show that the function $\Phi _ { f } : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R }$ is well defined and continuous.

We consider a function $f \in \mathscr { C } _ { b } ^ { 0 }$. We define for $( t , x ) \in \mathbb { R } ^ { 2 }$ $$\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$$ where $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$, and $Lf(x) = \frac{1}{2} f''(x) - x f'(x)$. We assume that $f \in \mathscr { C } _ { b } ^ { 2 }$.
5a. Show that, on $\mathbb { R } ^ { 2 } , \Phi _ { f }$ is of class $\mathscr { C } ^ { 1 }$ and $\partial _ { x x } \Phi _ { f }$ is well defined, continuous and bounded.
5b. Let $( t , x ) \in \mathbb { R } ^ { 2 }$. Find a relation between $\partial _ { x } \Phi _ { f } ( t , x )$ and $\Phi _ { f ^ { \prime } } ( t , x )$.
5c. Show that for all $( t , x ) \in \mathbb { R } ^ { 2 }$, we have $\partial _ { t } \Phi _ { f } ( t , x ) \cos t = L \Phi _ { f } ( t , x ) \sin t$.
5d. Show that for all $t \in \mathbb { R }$, we have $\int \Phi _ { f } ( t , x ) \mu ( x ) d x = \int f ( x ) \mu ( x ) d x$.

Let $f : \mathbb { R } \rightarrow \mathbb { R } _ { + }$ be a positive function in $\mathscr { C } _ { b } ^ { 0 }$. We define for $t \in \mathbb { R }$ $$J ( t ) = \int h \left( \Phi _ { f } ( t , x ) \right) \mu ( x ) d x$$ where $\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$, $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$, and $h(x) = x\ln(x)$ for $x > 0$, $h(0) = 0$. Show that $J : \mathbb { R } \rightarrow \mathbb { R }$ is continuous, and calculate $J ( 0 )$ and $J \left( \frac { \pi } { 2 } \right)$.

Let $f : \mathbb { R } \rightarrow \mathbb { R } _ { + }$ be a positive function in $\mathscr { C } _ { b } ^ { 0 }$. We define for $t \in \mathbb { R }$ $$J ( t ) = \int h \left( \Phi _ { f } ( t , x ) \right) \mu ( x ) d x$$ where $\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$, $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$, and $h(x) = x\ln(x)$ for $x > 0$, $h(0) = 0$. We assume throughout this question that $f \in \mathscr { C } _ { b } ^ { 2 }$ and that there exists $\delta > 0$ such that $$\forall x \in \mathbb { R } , \quad f ( x ) \geqslant \delta .$$ We denote $g = \left( f ^ { \prime } \right) ^ { 2 } / f$.
7a. Show that $J$ is then of class $\mathscr { C } ^ { 1 }$ on $\mathbb { R }$ and that $$\forall t \in \mathbb { R } , \quad J ^ { \prime } ( t ) \cos t = - \frac { \sin t } { 2 } \int \frac { \left( \partial _ { x } \Phi _ { f } ( t , x ) \right) ^ { 2 } } { \Phi _ { f } ( t , x ) } \mu ( x ) d x$$
7b. Let $( t , x ) \in \mathbb { R } ^ { 2 }$. Show that $$\Phi _ { f ^ { \prime } } ( t , x ) ^ { 2 } \leqslant \Phi _ { f } ( t , x ) \Phi _ { g } ( t , x )$$
7c. Conclude that $$\int h ( f ( x ) ) \mu ( x ) d x - h \left( \int f ( y ) \mu ( y ) d y \right) \leqslant \frac { 1 } { 4 } \int g ( x ) \mu ( x ) d x$$

We define $\hat{f}$ on $\mathbb{R}_{+}^{*} \times \mathbb{R}$ by: $\forall(t, \xi) \in \mathbb{R}_{+}^{*} \times \mathbb{R},\ \hat{f}(t, \xi) = \int_{-\infty}^{+\infty} f(t, x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x$. Show that, for any real number $\xi$, $\lim_{t \rightarrow 0^{+}} \hat{f}(t, \xi) = \widehat{g_{\sigma}}(\xi)$. One may use any sequence $\left(t_{n}\right)_{n \in \mathbb{N}}$ of strictly positive reals converging to zero.

Show that, for any real $\xi$ and any real $t > 0$, $\frac{\partial \hat{f}}{\partial t}(t, \xi) = \int_{-\infty}^{+\infty} \frac{\partial f}{\partial t}(t, x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x$.

By noting that $\int_{-\infty}^{+\infty} \frac{\partial f}{\partial t}(t, x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x = \mathcal{F}\left(\frac{\partial f}{\partial t}(t, \cdot)\right)(\xi)$ and using question 7, show that, for any real $\xi$ and any real $t > 0$, $\frac{\partial \hat{f}}{\partial t}(t, \xi) = -4\pi^{2} \xi^{2} \hat{f}(t, \xi)$.

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 assume here that $\lambda = 0$. Deduce from this, in the case $\lambda = 0$, the harmonic functions with separable polar variables.

Show that, for all $\xi \in \mathbb{R}$, there exists a real $K(\xi)$ such that for all $t \in \mathbb{R}_{+}^{*}$, $\hat{f}(t, \xi) = K(\xi) \exp\left(-4\pi^{2} \xi^{2} t\right)$.

Using question 15, determine, for any real $\xi$, the value of $K(\xi)$.

Let $t$ be a strictly positive real number. Using questions 20 and 12, and the result that if $u$ and $v$ are functions from $\mathbb{R}$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}$ and satisfying $\mathcal{F}(u) = \mathcal{F}(v)$, then $u = v$, deduce the existence of a real $\lambda_{t,\sigma}$ such that $$f(t, \cdot) = \lambda_{t,\sigma} g_{\sqrt{\sigma^{2}+2t}}$$

Show that the function $I : \left\lvert\, \begin{aligned} & \mathbb{R}_{+}^{*} \rightarrow \mathbb{R} \\ & t \mapsto \int_{-\infty}^{+\infty} f(t, x) \mathrm{d}x \end{aligned}\right.$ is constant. One may use the result of question 17.

Let $t \in \mathbb{R}_{+}^{*}$ and $x \in\ ]0,1[$. Give the limit, as $\theta$ tends to zero, of $\frac{f(t+\theta, x) - f(t, x)}{\theta}$.

Let $t \in \mathbb{R}_{+}^{*}$ and $x \in\ ]0,1[$. Show that $\lim_{h \rightarrow 0} \frac{f(t, x+h) - 2f(t, x) + f(t, x-h)}{h^{2}} = \frac{\partial^{2} f}{\partial x^{2}}(x, t)$.

Let $\tau$ be a strictly positive real and $q$ a natural integer greater than or equal to 2. We set $\delta = \frac{1}{q+1}$ and $r = \frac{\tau}{\delta^{2}}$. The numerical scheme imposes, for any natural integer $n$ and any $k \in \llbracket 1, q \rrbracket$: $$\frac{f_{n+1}(k) - f_{n}(k)}{\tau} = \frac{f_{n}(k+1) - 2f_{n}(k) + f_{n}(k-1)}{\delta^{2}}$$ as well as $f_{n}(0) = f_{n}(q+1) = 0$. We set $F_{n} = \left(\begin{array}{c} f_{n}(1) \\ \vdots \\ f_{n}(q) \end{array}\right)$, $I_{q}$ is the identity matrix of order $q$, $B$ is the square matrix of order $q$ with coefficient $(i,j)$ equal to 1 if $|i-j|=1$ and 0 otherwise, and $A = (1-2r)I_{q} + rB$. Show that, for all $n \in \mathbb{N}$, $F_{n+1} = A F_{n}$.

We define a sequence $(a_n)_{n \geqslant 1}$ by setting $$\forall n \in \mathbb{N}^*, \quad a_n = \frac{(-n)^{n-1}}{n!}.$$ We define, when possible, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$, with radius of convergence $R$. We consider the function $$h : \begin{array}{ccc} ]-R,R[ & \rightarrow & \mathbb{R} \\ x & \mapsto & S(x)\mathrm{e}^{S(x)} \end{array}$$ Prove that $h$ is a solution on $]-R, R[$ of the differential equation $xy' - y = 0$.

Solve the differential equation $xy' - y = 0$ on each of the intervals $]0, R[$ and $]-R, 0[$ then on the interval $]-R, R[$.

For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ and a sequence of functions $(w_n)_{n \geqslant 0}$ on $\mathbb{R}^+$ defined by $$\forall x \in \mathbb{R}^+, \quad \begin{cases} w_0(x) = 1 \\ w_{n+1}(x) = \phi_x(w_n(x)) \end{cases}$$ Let $W$ be the Lambert function defined in Part I. Prove that, for every positive real $x$, $W(x)$ is a fixed point of $\phi_x$, that is, a solution of the equation $\phi_x(t) = t$.

For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ and a sequence of functions $(w_n)_{n \geqslant 0}$ on $\mathbb{R}^+$ defined by $$\forall x \in \mathbb{R}^+, \quad \begin{cases} w_0(x) = 1 \\ w_{n+1}(x) = \phi_x(w_n(x)) \end{cases}$$ Let $W$ be the Lambert function defined in Part I. Using the fact that $0 \leqslant \phi_x'(t) \leqslant \frac{x}{\mathrm{e}}$ for all $t \in \mathbb{R}$, deduce that $$\forall x \in [0, \mathrm{e}], \quad \forall n \in \mathbb{N}, \quad |w_n(x) - W(x)| \leqslant \left(\frac{x}{\mathrm{e}}\right)^n |1 - W(x)|.$$

For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ and a sequence of functions $(w_n)_{n \geqslant 0}$ on $\mathbb{R}^+$ defined by $$\forall x \in \mathbb{R}^+, \quad \begin{cases} w_0(x) = 1 \\ w_{n+1}(x) = \phi_x(w_n(x)) \end{cases}$$ Let $W$ be the Lambert function defined in Part I. For every real $a \in ]0, \mathrm{e}[$, justify that the sequence of functions $(w_n)$ converges uniformly on $[0, a]$ to the function $W$.