We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set $$r _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x }, \quad R _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t$$
Show that the remainder is of the order of the first neglected term, that is, for all $N \geqslant 1$, $$R _ { N } ( x ) \sim r _ { N } ( x ) \quad \text { when } \quad x \rightarrow 0$$

We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set $$r _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x }$$
Show that, for $0 < x < 1 / 2$, the sequence $\left( \left| r _ { N } ( x ) \right| \right) _ { N \geqslant 1 }$ is decreasing up to a certain rank, then increasing.

We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set $$\begin{aligned} r _ { N } ( x ) & = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x } \\ S _ { N } ( x ) & = \sum _ { k = 1 } ^ { N } ( - 1 ) ^ { k - 1 } ( k - 1 ) ! x ^ { k } e ^ { - 1 / x } \\ R _ { N } ( x ) & = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t \end{aligned}$$ The relative error is $E _ { N } ( x ) = \left| \frac { R _ { N } ( x ) } { F ( x ) } \right|$.
Show that, if $N$ is even: $N = 2 M$ with $M \geqslant 1$, and if $0 < x \leqslant 1 / N$, we have $S _ { N } ( x ) \geqslant 0$ and $$E _ { N } ( x ) \leqslant \frac { N ! x ^ { N + 1 } } { \sum _ { \ell = 0 } ^ { M - 1 } ( 1 - ( 2 \ell + 1 ) x ) ( 2 \ell ) ! x ^ { 2 \ell + 1 } }$$

We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set $$\begin{aligned} S _ { N } ( x ) & = \sum _ { k = 1 } ^ { N } ( - 1 ) ^ { k - 1 } ( k - 1 ) ! x ^ { k } e ^ { - 1 / x } \\ R _ { N } ( x ) & = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t \end{aligned}$$ The relative error is $E _ { N } ( x ) = \left| \frac { R _ { N } ( x ) } { F ( x ) } \right|$.
Verify that $E _ { 4 } \left( \frac { 1 } { 10 } \right) \leqslant 3.10 ^ { - 3 }$.

Let $\left(a_{k}\right)_{k \in \mathbb{N}}$ be a sequence of complex terms such that the series with general term $k^{2} a_{k}$ is absolutely convergent. For $(t, x) \in \mathbb{R}^{2}$ we then denote $$\Phi_{0}(t, x) = \sum_{k=0}^{+\infty} a_{k} e^{-ik^{2}t + ikx}$$
(a) Show that $\Phi_{0}$ is well defined on $\mathbb{R}^{2}$.
(b) Show that for all $(t, x) \in \mathbb{R}^{2}$ we have $\Phi_{0}(., x) \in C^{1}(\mathbb{R}, \mathbb{C})$ and $\Phi_{0}(t, .) \in C^{2}(\mathbb{R})$. Calculate $\frac{\partial \Phi_{0}}{\partial t}(t, x)$ and $\frac{\partial^{2} \Phi_{0}}{\partial x^{2}}(t, x)$.
(c) Let $(c_{k})$ be a sequence of complex terms such that the series with general term $k^{2} c_{k}$ is absolutely convergent. For $x \in \mathbb{R}$, we set $$f_{0}(x) = \sum_{k=0}^{+\infty} c_{k} e^{ikx}$$ Construct a function $\Psi_{0}$ defined on $\mathbb{R}_{+}^{*} \times \mathbb{R}$ which satisfies

• For all $(t, x) \in \mathbb{R}_{+}^{*}$, $\Psi_{0}(., x) \in C^{1}(\mathbb{R}_{+}^{*}, \mathbb{C})$ and $\Psi_{0}(t, .) \in C^{2}(\mathbb{R}, \mathbb{C})$ and $\Psi_{0}$ is a solution of equation $(F_{0})$, that is $$\forall (t, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R}, \quad i\frac{\partial \Psi_{0}}{\partial t}(t, x) + \frac{\partial^{2} \Psi_{0}}{\partial x^{2}}(t, x) + \frac{1}{2t}\Psi_{0}(t, x) = 0$$
• For all $t > 0$, $\Psi_{0}(t, .)$ is periodic.
• For all $x \in \mathbb{R}$, $\Psi_{0}(1, x) = f_{0}(x)$.

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Deduce the variations of $\Gamma$ on $\mathcal{D}$. Specify in particular the limits of $\Gamma$ at 0 and at $+\infty$. Also specify the limits of $\Gamma^{\prime}$ at 0 and at $+\infty$. Sketch the graph of $\Gamma$.

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$. Show that in a neighbourhood of $x = 0$, the function $F$ can be written in the form
$$F(x) = \sum_{n=0}^{+\infty} c_{n} \frac{(\mathrm{i}x)^{n}}{n!} \tag{S}$$
where $c_{n}$ is the value of Gamma at a point to be specified. Express $c_{n}$ in terms of $n$ and $c_{0}$.
What is the radius of convergence of the power series appearing on the right-hand side of $(S)$?

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 $$F(x) = \sum_{n=0}^{+\infty} c_{n} \frac{(\mathrm{i}x)^{n}}{n!} \tag{S}$$ We admit that $\Gamma(x) \underset{x \rightarrow +\infty}{\sim} \sqrt{2\pi} x^{(x-1/2)} \mathrm{e}^{-x}$.
Investigate whether the series on the right-hand side of $(S)$ converges absolutely when $|x| = R$, where $R$ is the radius of convergence.

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$. Let $R(x)$ be the real part and $I(x)$ be the imaginary part of $F(x)$.
Determine, in a neighbourhood of 0, the Taylor expansion of $R(x)$ to order 3 and of $I(x)$ to order 4.

We consider a real $\alpha$ such that for every prime number $p$, $p^\alpha$ is a natural number. We apply relation $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f(x+j) = f^{(n)}(x + y_n) \quad \text{(IV.1)}$$ to the function $f_\alpha(x) = x^\alpha$ and to the integer $n = \lfloor \alpha \rfloor + 1$. The notations are those of question IV.A.4.
What is the limit of the expression $f_\alpha^{(n)}(x + y_n)$ when $x \in \mathbb{N}^*$ tends to $+\infty$?

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

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 $f \in \mathcal{S}$. We assume that $\mathcal{F}(f)$ is integrable on $\mathbb{R}$. For every positive natural number $n$, we set
$$I_{n} = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \theta\left(\frac{\xi}{n}\right) \mathrm{d}\xi \quad J_{n} = \int_{-\infty}^{+\infty} f\left(\frac{t}{n}\right) \mathcal{F}(\theta)(t) \mathrm{d}t$$
Show that $\lim_{n \rightarrow +\infty} I_{n} = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \mathrm{d}\xi$.

Let $f \in \mathcal{S}$. We assume that $\mathcal{F}(f)$ is integrable on $\mathbb{R}$. For every positive natural number $n$, we set
$$I_{n} = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \theta\left(\frac{\xi}{n}\right) \mathrm{d}\xi \quad J_{n} = \int_{-\infty}^{+\infty} f\left(\frac{t}{n}\right) \mathcal{F}(\theta)(t) \mathrm{d}t$$
Calculate $\lim_{n \rightarrow +\infty} J_{n}$.

Let $f \in \mathcal{S}$. We assume that $\mathcal{F}(f)$ is integrable on $\mathbb{R}$. For every positive natural number $n$, we set
$$I_{n} = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \theta\left(\frac{\xi}{n}\right) \mathrm{d}\xi \quad J_{n} = \int_{-\infty}^{+\infty} f\left(\frac{t}{n}\right) \mathcal{F}(\theta)(t) \mathrm{d}t$$
Prove that $\forall n \in \mathbb{N}^{*}, I_{n} = J_{n}$.
We will admit the Fubini formula:
$$\int_{-\infty}^{+\infty} \left(\int_{-\infty}^{+\infty} f(t) \theta\left(\frac{\xi}{n}\right) e^{-2\pi\mathrm{i} t\xi} \mathrm{d}\xi\right) \mathrm{d}t = \int_{-\infty}^{+\infty} \left(\int_{-\infty}^{+\infty} f(t) \theta\left(\frac{\xi}{n}\right) e^{-2\pi\mathrm{i} t\xi} \mathrm{d}t\right) \mathrm{d}\xi$$

Let $f \in \mathcal{S}$. We assume that $\mathcal{F}(f)$ is integrable on $\mathbb{R}$.
Prove that $f(0) = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \mathrm{d}\xi$.
Deduce, using the function $h : t \mapsto f(x+t)$, that
$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$

Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. The sequence of complex numbers $(c_{n}(f))_{n \in \mathbb{Z}}$ is defined by
$$\forall n \in \mathbb{Z}, \quad c_{n}(f) = \int_{-1/2}^{1/2} f(x) e^{-2\pi\mathrm{i} nx} \mathrm{d}x$$
Prove the existence of a real number $E$ such that
$$\forall t \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad \left|f(t) - \sum_{k=-n}^{n} c_{k}(f) e^{2\pi\mathrm{i} kt}\right| \leqslant \frac{E}{2n+1}$$
One may introduce the function $h_{t} : x \mapsto f(x+t)$.

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 $[-1/2, 1/2]$. Show that $h$ is of class $C^{\infty}$ on $\mathbb{R}$.

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 recall that the hyperbolic cosine function, which we denote cosh, is defined, for every real $t$, by $$\cosh(t)=\frac{\mathrm{e}^{t}+\mathrm{e}^{-t}}{2}$$
a) Give the power series expansion of the hyperbolic cosine function and that of the function defined on $\mathbb{R}$ by $t \mapsto \mathrm{e}^{t^{2}/2}$. We will give the radius of convergence of these two power series.
b) Deduce that $\forall t \in \mathbb{R}, \cosh(t) \leqslant \mathrm{e}^{t^{2}/2}$.

We assume that $f \in \mathcal { C } ( [ 0,1 ] , \mathbb { R } )$ satisfies: $$\exists \alpha \in ]0,1] , \exists K \geq 0 , \forall ( y , z ) \in [ 0,1 ] ^ { 2 } , | f ( y ) - f ( z ) | \leq K | y - z | ^ { \alpha }$$ and that $c(x) = 0$ for all $x \in [0,1]$. For all $n \in \mathbb{N}^*$, we define: $$B _ { n } f ( X ) = \sum _ { k = 0 } ^ { n } f \left( \frac { k } { n } \right) \binom { n } { k } X ^ { k } ( 1 - X ) ^ { n - k }$$
Let $x \in ]0,1[$ and $n \in \mathbb { N } ^ { * }$. We consider $X _ { 1 } , \ldots , X _ { n }$ mutually independent random variables all following the same Bernoulli distribution with parameter $x$. We set
$$S _ { n } = \frac { X _ { 1 } + \cdots + X _ { n } } { n }$$
(a) Express $\mathbb { E } \left( S _ { n } \right) , \mathbb { V } \left( S _ { n } \right)$ and $\mathbb { E } \left( f \left( S _ { n } \right) \right)$ in terms of $x , n$ and the polynomial $B _ { n } f$.
(b) Deduce the inequalities:
$$\sum _ { k = 0 } ^ { n } \left| x - \frac { k } { n } \right| \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k } \leq \mathbb { V } \left( S _ { n } \right) ^ { \frac { 1 } { 2 } } \leq \frac { 1 } { 2 \sqrt { n } }$$