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 $\forall t \in \mathbb{R}, \cosh(t) \leqslant \mathrm{e}^{t^{2}/2}$.

Using the results of the previous questions (in particular the integral representation of $I_n$ from question 2, the bounds from question 3, and the Gaussian integral $\int_{-\infty}^{+\infty} e^{-x^{2}/2}\, dx = \sqrt{2\pi}$), deduce Stirling's formula: $$n! \underset{n \rightarrow \infty}{\sim} \sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}.$$

We assume that, for all non-zero natural integer $n$, $X$ admits a moment of order $n$ and that the power series $\sum _ { n \geqslant 0 } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$ has a radius of convergence $R _ { X } > 0$. For all $t \in ] - R _ { X } , R _ { X } [$, we denote $M _ { X } ( t ) = \sum _ { n = 0 } ^ { + \infty } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$.
Justify that knowledge of the function $M _ { X }$ allows us to determine uniquely the sequence $\left( m _ { n } ( X ) \right) _ { n \in \mathbb { N } ^ { * } }$.

Show that, for any real $\xi$, there exists a real sequence $\left(c_{p}(\xi)\right)_{p \in \mathbb{N}}$ such that $$\forall x \in \mathbb{R}, \quad \exp\left(-x^{2}\right) \cos(2\pi \xi x) = \sum_{p=0}^{+\infty} c_{p}(\xi) \exp\left(-x^{2}\right) x^{2p}$$

Let $n \in \mathbb{N}^*$, $h = \frac{1}{n+1}$, and $x_i = ih$ for all $i \in \{0, \ldots, n+1\}$. Show that for any function $v \in \mathcal { C } ^ { 4 } ( [ 0,1 ] , \mathbb { R } )$, there exists a constant $C \geq 0$, independent of $n$, such that
$$\forall i \in \{ 1 , \ldots , n \} , \left| v ^ { \prime \prime } \left( x _ { i } \right) - \frac { 1 } { h ^ { 2 } } \left( v \left( x _ { i + 1 } \right) + v \left( x _ { i - 1 } \right) - 2 v \left( x _ { i } \right) \right) \right| \leq C h ^ { 2 }$$

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 $\varphi ^ { ( p ) } ( 1 )$.

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

We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Demonstrate, for all $x \in ] - 1,1 [$, $$\varphi ( x ) = \sum _ { q = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { q } } { q ! } x ^ { q } ( 1 - x ) ^ { - q / 2 }$$

Deduce the existence of a real $\nu_{\sigma}$ such that, for any real $\xi$ and any real $t > 0$, $$\hat{f}(t, \xi) = \nu_{\sigma} \exp\left(-2\pi^{2}\left(\sigma^{2}+2t\right) \xi^{2}\right)$$

Give the value of $\nu_{\sigma}$.

A function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands as 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 as 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 as 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 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 $D(0,R)$.

A function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands as 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 $D(0,R)$.

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 as 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 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 $D(0,R)$.

Show that if $g$ belongs to $\mathcal{H}(D(0,R))$ then there exists a function $H$ that expands as 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.

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.

Let $f$ be a function that expands as a power series on $D(0,R)$, i.e., there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i}y)^n$$ Show that for all $r \in [0, R[$, we have $f(0) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos(t), r\sin(t))\, \mathrm{d}t$.

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 for all $r \in [0, R[$, we have $f(0) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos(t), r\sin(t)) \, \mathrm{d}t$.

Show an analogous result to $f(0) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos(t), r\sin(t))\, \mathrm{d}t$ for harmonic functions.

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