We consider the function $\psi$ defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}^{*}, \quad \psi(x) = \frac{\sin(\pi x)}{\pi x} \quad \text{and} \quad \psi(0) = 1$$
I.B.1) Justify that $\psi$ is expandable as a power series. Specify this expansion and its radius of convergence. Deduce that $\psi$ is of class $C^{\infty}$ on $\mathbb{R}$.
I.B.2) Prove that
$$\forall n \in \mathbb{N}, \quad \int_{n}^{n+1} |\psi(x)| \mathrm{d}x \geqslant \frac{2}{(n+1)\pi^{2}}$$
Deduce that $\psi$ does not belong to $E_{\mathrm{cpm}}$.

Let $f \in \mathcal{S}$.
I.D.1) Justify that, for every natural number $n$, the function $x \mapsto x^{n} f(x)$ is integrable on $\mathbb{R}$.
I.D.2) Prove that the function $\mathcal{F}(f)$ is of class $C^{\infty}$ on $\mathbb{R}$ and that
$$\forall n \in \mathbb{N}, \quad \forall \xi \in \mathbb{R}, \quad (\mathcal{F}(f))^{(n)}(\xi) = (-2\pi\mathrm{i})^{n} \int_{-\infty}^{+\infty} t^{n} f(t) e^{-2\pi\mathrm{i} t\xi} \mathrm{~d}t$$

Let $f$ be a function in $\mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. According to the Fourier inversion formula, we have
$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-1/2}^{1/2} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$
Prove that $\mathcal{F}(f)$ is of class $C^{\infty}$ on $\mathbb{R}$ and that $\mathcal{F}(f) \in \mathcal{S}$. Deduce that $f$ is of class $C^{\infty}$ on $\mathbb{R}$.

Let $f$ be a function in $\mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. According to the Fourier inversion formula, we have
$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-1/2}^{1/2} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$
Prove that
$$\forall (x, x_{0}) \in \mathbb{R}^{2}, \quad \sum_{n=0}^{+\infty} \frac{(x-x_{0})^{n}}{n!} \int_{-1/2}^{1/2} (2\pi\mathrm{i}\xi)^{n} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x_{0}\xi} \mathrm{d}\xi = f(x)$$

Let $f$ be a function in $\mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. According to the Fourier inversion formula, we have
$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-1/2}^{1/2} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$
We assume that $f$ is zero outside a segment $[a, b]$. Show that $f = 0$.

Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. We set
$$\forall k \in \mathbb{Z}, \quad \forall x \in \mathbb{R}, \quad \psi_{k}(x) = \psi(x+k)$$
where $\psi(x) = \frac{\sin(\pi x)}{\pi x}$ for $x \neq 0$ and $\psi(0) = 1$.
Justify that $\forall n \in \mathbb{N}, \quad (\mathcal{F}(f))^{(n)}\left(\frac{1}{2}\right) = (\mathcal{F}(f))^{(n)}\left(-\frac{1}{2}\right) = 0$.

Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. Let $h$ be the function defined on $\mathbb{R}$, which is 1-periodic and which equals $\mathcal{F}(f)$ on the interval $[-1/2, 1/2]$.
Using the inequality from IV.G, prove the existence of a sequence of complex numbers $(d_{k})_{k \in \mathbb{Z}}$ such that the sequence of functions $\left(x \mapsto \sum_{k=-n}^{n} d_{k} e^{2\pi\mathrm{i} kx}\right)_{n \in \mathbb{N}}$ converges uniformly to $\mathcal{F}(f)$ on $[-1/2, 1/2]$.

Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. We set
$$\forall k \in \mathbb{Z}, \quad \forall x \in \mathbb{R}, \quad \psi_{k}(x) = \psi(x+k)$$
where $\psi(x) = \frac{\sin(\pi x)}{\pi x}$ for $x \neq 0$ and $\psi(0) = 1$.
Let $(d_{k})_{k \in \mathbb{Z}}$ be the sequence of complex numbers from V.C. Prove that the sequence of functions $\left(\sum_{k=-n}^{n} d_{k} \psi_{k}\right)_{n \in \mathbb{N}}$ converges uniformly to $f$ on $\mathbb{R}$.

Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. We set
$$\forall k \in \mathbb{Z}, \quad \forall x \in \mathbb{R}, \quad \psi_{k}(x) = \psi(x+k)$$
where $\psi(x) = \frac{\sin(\pi x)}{\pi x}$ for $x \neq 0$ and $\psi(0) = 1$, and $f = \sum_{k=-\infty}^{+\infty} d_{k} \psi_{k}$ (uniform limit).
Establish that $\forall j \in \mathbb{Z},\ f(-j) = d_{j}$.

We say that a function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands in a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ We admit the following result: a function $h$ from $D(0,R)$ to $\mathbb{C}$ expands in a power series on $D(0,R)$ if and only if $h$ is of class $\mathcal{C}^1$ on $D(0,R)$ and for all $(x,y) \in D(0,R)$, $\frac{\partial h}{\partial y}(x,y) = \mathrm{i} \frac{\partial h}{\partial x}(x,y)$.
Show that if $f$ does not vanish on $D(0,R)$ then $1/f$ expands in a power series on $D(0,R)$.

We say that a function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands in a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ We denote by $u$ and $v$ the real and imaginary parts of $f$. Show that the function $uv$ is harmonic on $D(0,R)$.

Let $g$ be a function from $D(0,R) \subset \mathbb{R}^2$ to $\mathbb{R}$. We assume that $g$ is harmonic. Show that the function $h$ defined on $D(0,R)$ by $$h : (x,y) \longmapsto \frac{\partial g}{\partial x}(x,y) - \mathrm{i} \frac{\partial g}{\partial y}(x,y)$$ expands in a power series on $D(0,R)$.

Let $g$ be a function from $D(0,R) \subset \mathbb{R}^2$ to $\mathbb{R}$. We assume that $g$ is harmonic. Show that if $g$ belongs to $\mathcal{H}(D(0,R))$ then there exists a function $H$ that expands in a power series on $D(0,R)$ such that $g$ is the real part of $H$.
One may consider a power series that is a primitive of the power series associated with the function $h$ from the previous question.

Show an analogous result to Q32 for harmonic functions: for a harmonic function $g$ on $D(0,R)$, show that for all $r \in [0, R[$, $g(0) = \frac{1}{2\pi} \int_0^{2\pi} g(r\cos(t), r\sin(t)) \, \mathrm{d}t$.

Throughout this part, $f$ denotes a function that expands in a power series on $D(0,R)$, i.e., $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ Show that $\forall r \in [0, R[$, $|f(0)| \leqslant \sup_{t \in \mathbb{R}} |f(r\cos(t), r\sin(t))|$.

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

We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Deduce that $\varphi$ is of class $C ^ { \infty }$ on $\mathbb { R }$ and for $p \in \mathbb { N } ^ { * }$, give the value of $\varphi ^ { ( p ) } ( 1 )$.

We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by $$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$ The purpose of Part III is to construct a function of class $C ^ { \infty }$ on $\mathbb { R }$, non-zero, whose all moments of order $p$ ($p \in \mathbb { N }$) are zero. Using the results of questions 36 and 37, conclude.

Let $f(x) = \frac{\sin x + 1}{\cos x}$ on $I = ]-\pi/2, \pi/2[$ and $g$ the sum of the power series $\sum_{n \in \mathbb{N}} \frac{\alpha_n}{n!} x^n$. Both satisfy $2h^{\prime}(x) = h(x)^2 + 1$. By considering the functions $\arctan f$ and $\arctan g$, show $$\forall x \in I, \quad f(x) = g(x).$$

If $a$ and $b$ are two real numbers, we denote $K _ { a , b }$ the function defined for all real $t$ by $K _ { a , b } ( t ) = \begin{cases} \frac { \mathrm { e } ^ { \mathrm { i } t b } - \mathrm { e } ^ { \mathrm { i } t a } } { 2 \mathrm { i } t } & \text { if } t \neq 0 , \\ \frac { b - a } { 2 } & \text { if } t = 0 . \end{cases}$ Using power series, show that $K _ { a , b }$ is of class $C ^ { \infty }$ on $\mathbb { R }$.

Let $\Gamma$ be the pointwise limit on $]0, +\infty[$ of the sequence $\left(\Gamma_n\right)_{n \geqslant 1}$ where $$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$ Let $f : ]0, +\infty[ \rightarrow ]0, +\infty[$ be a function of class $\mathscr{C}^2$ such that the function $\ln(f)$ is convex and satisfies $f(1) = 1$ and $f(x+1) = xf(x)$ for all $x > 0$.
Show that the function $$S : \begin{array}{ccc} ]0, +\infty[ & \longrightarrow & \mathbb{R} \\ x & \longmapsto & \ln\left(\frac{f(x)}{\Gamma(x)}\right) \end{array}$$ is 1-periodic and convex.

Let $\Gamma$ be the pointwise limit on $]0, +\infty[$ of the sequence $\left(\Gamma_n\right)_{n \geqslant 1}$ where $$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$ Let $f : ]0, +\infty[ \rightarrow ]0, +\infty[$ be a function of class $\mathscr{C}^2$ such that the function $\ln(f)$ is convex and satisfies $f(1) = 1$ and $f(x+1) = xf(x)$ for all $x > 0$. The function $S(x) = \ln\left(\frac{f(x)}{\Gamma(x)}\right)$ is 1-periodic and convex.
Deduce that $f = \Gamma$.

Using the result that for $x \in ]-\frac{1}{2}, \frac{1}{2}[$: $$\frac{\pi}{\cos(\pi x)} = \sum_{k=0}^{+\infty} \left(\sum_{n=0}^{+\infty} \frac{(-1)^n}{(2n+1)^{2k+1}}\right) 2^{2k+2} x^{2k},$$ deduce that the function $$v : \begin{array}{ccc} ]-\frac{\pi}{2}, \frac{\pi}{2}[ & \longrightarrow & \mathbb{R} \\ x & \longmapsto & \frac{1}{\cos(x)} \end{array}$$ is expandable as a power series and that, for all $k \in \mathbb{N}$, $$\sum_{n=0}^{+\infty} \frac{(-1)^n}{(2n+1)^{2k+1}} = \frac{\pi^{2k+1}}{2^{2k+2}(2k)!} E_{2k}$$ where, for all $k \in \mathbb{N}$, $E_{2k} = v^{(2k)}(0)$.

Let $a > 0$, $I = [-a, a]$, and $$f : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ x & \mapsto & \dfrac{1}{1+x^2} \end{array}.$$ Show that $f$ is of class $\mathcal { C } ^ { \infty }$ and that, for all $k$ in $\mathbb { N }$ and all $t \in \left] - \pi / 2 , \pi / 2 \right[$, $$f ^ { ( k ) } ( \tan t ) = k ! \cos ^ { k + 1 } ( t ) \cos ( ( k + 1 ) t + k \pi / 2 ).$$

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 $\mathbb { R }$.