For $(x,y) \in D(0,1)$ fixed, we define the complex number $z = x + iy$ and we set for $t$ real: $$\mathrm{N}(x,y,t) = \frac{1 - |z|^2}{|z - e^{it}|^2} = \frac{1 - (x^2 + y^2)}{(x - \cos t)^2 + (y - \sin t)^2}$$
Deduce that $\dfrac{1}{2\pi} \int_0^{2\pi} \mathrm{N}(x,y,t)\, \mathrm{d}t = 1$.
One may write $\dfrac{1}{1 - ze^{-it}}$ in the form of the sum of a series of functions.

For every pair of natural integers $( p , q )$ and for every $\varepsilon \in ] 0,1 [$, we denote $$I _ { p , q } = \int _ { 0 } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t \quad \text { and } \quad I _ { p , q } ^ { \varepsilon } = \int _ { \varepsilon } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t$$
Show that the integral $I _ { p , q }$ exists for every pair of natural integers $( p , q )$.

Let $x > 0$. Show that $t \mapsto t ^ { x - 1 } e ^ { - t }$ is integrable on $] 0 , + \infty [$.

Throughout the rest of the problem, we denote $\Gamma$ the function defined on $\mathbb { R } ^ { + * }$ by $\Gamma ( x ) = \int _ { 0 } ^ { + \infty } t ^ { x - 1 } e ^ { - t } \mathrm {~d} t$. We admit that $\Gamma$ is of class $\mathcal { C } ^ { \infty }$ on its domain of definition, takes strictly positive values and satisfies, for every real $x > 0$, the relation $\Gamma ( x + 1 ) = x \Gamma ( x )$.
Let $x$ and $\alpha$ be two strictly positive real numbers. Justify the existence of $\int _ { 0 } ^ { + \infty } t ^ { x - 1 } e ^ { - \alpha t } \mathrm {~d} t$ and give its value as a function of $\Gamma ( x )$ and $\alpha ^ { x }$.

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$.
Justify the existence of $\beta ( x , y )$ for $x > 0$ and $y > 0$.

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$.
Show that for all real $x > 0$ and $y > 0 , \beta ( x , y ) = \beta ( y , x )$.

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

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
$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$

For every natural number $n$, we denote by $S_{n}$ the function defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad S_{n}(x) = \sum_{k=-n}^{n} e^{2\pi\mathrm{i} kx}$$
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
$$\forall n \in \mathbb{Z}, \quad c_{n}(f) = \int_{-1/2}^{1/2} f(x) e^{-2\pi\mathrm{i} nx} \mathrm{d}x$$
Let $n \in \mathbb{N}$. Calculate the integral $\int_{-1/2}^{1/2} S_{n}(x) \mathrm{d}x$.

For every natural number $n$, we denote by $S_{n}$ the function defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad S_{n}(x) = \sum_{k=-n}^{n} e^{2\pi\mathrm{i} kx}$$
Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. We consider the function $g$ defined on $[-1,1]$ by
$$\forall x \in ]-1,1[\backslash\{0\}, \quad g(x) = \frac{f(x)-f(0)}{\sin(\pi x)} \quad g(0) = \frac{f'(0)}{\pi} \quad g(1) = g(-1) = -g(0)$$
and the sequence of complex numbers $(c_{n}(f))_{n \in \mathbb{Z}}$ defined by
$$\forall n \in \mathbb{Z}, \quad c_{n}(f) = \int_{-1/2}^{1/2} f(x) e^{-2\pi\mathrm{i} nx} \mathrm{d}x$$
Justify that
$$\forall n \in \mathbb{N}^{*}, \quad \sum_{k=-n}^{n} c_{k}(f) = f(0) + \int_{-1/2}^{1/2} g(x) \sin((2n+1)\pi x) \mathrm{d}x$$

Determine for which $x \in \mathbb{R}$ the integral below is convergent $$\int_{0}^{1} \frac{t^{x} - 1}{1 - t} \mathrm{~d}t$$

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ By noting that, for all $t \in [0,1[$, $\frac{1}{1-t} = \sum_{n=0}^{\infty} t^{n}$, show that $$\forall x \in {]-1,+\infty[}, \quad f(x) = \int_{0}^{1} \frac{t^{x} - 1}{1 - t} \mathrm{~d}t$$

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, set $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z)\, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Let $\varphi \in \mathbb{R}$. Show that, for any complex number $z$ such that $|z| < 1$, $g(z) = \frac{1}{2\pi} \int_{\varphi}^{\varphi + 2\pi} h(t) \mathcal{P}(t,z)\, \mathrm{d}t$.

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Let $\varphi \in \mathbb{R}$. Show that, for any complex number $z$ such that $|z| < 1$, $g(z) = \frac{1}{2\pi} \int_{\varphi}^{\varphi + 2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t$.

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, set $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z)\, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that, for all $\delta \in ]0, \pi[$ and all real $\varphi$, $\int_{\varphi+\delta}^{\varphi+2\pi-\delta} \mathcal{P}(t,z)\, \mathrm{d}t \xrightarrow[z \to \mathrm{e}^{\mathrm{i}\varphi}]{} 0$.

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. 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 that, for all $\delta \in ]0, \pi[$ and all real $\varphi$, $$\int_{\varphi+\delta}^{\varphi+2\pi-\delta} \mathcal{P}(t,z) \, \mathrm{d}t \xrightarrow[z \rightarrow \mathrm{e}^{\mathrm{i}\varphi}]{} 0$$

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, set $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z)\, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Using Heine's theorem, show that, for all $\varepsilon > 0$, there exists $\delta > 0$ such that, for all real number $\varphi$ and all complex number $z$ satisfying $|z| < 1$, $$|g(z) - h(\varphi)| \leqslant \frac{\sup_{t \in \mathbb{R}} |h(t)|}{\pi} \int_{\varphi+\delta}^{\varphi+2\pi-\delta} \mathcal{P}(t,z)\, \mathrm{d}t + \varepsilon$$

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Using Heine's theorem, show that, for all $\varepsilon > 0$, there exists $\delta > 0$ such that, for all real number $\varphi$ and all complex number $z$ satisfying $|z| < 1$, $$|g(z) - h(\varphi)| \leqslant \frac{\sup_{t \in \mathbb{R}} |h(t)|}{\pi} \int_{\varphi+\delta}^{\varphi+2\pi-\delta} \mathcal{P}(t,z) \, \mathrm{d}t + \varepsilon$$

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.

Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}$.
Justify the existence of $\int_0^1 \frac{t-1}{\ln t} \,\mathrm{d}t$ and then determine its value.
One may consider $\int_0^1 \mathbb{E}(t^{Y_n}) \,\mathrm{d}t$.

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}.$$ For every continuous function $f$ from $[0,1]$ to $\mathbb{R}$, the sequence $(\mathbb{E}(f(Y_n)))_{n \geqslant 1}$ converges to $\int_0^1 f(t)\,\mathrm{d}t$.
An application. Justify the existence of $\int_0^1 \frac{t-1}{\ln t}\,\mathrm{d}t$ then determine its value.
One may consider $\int_0^1 \mathbb{E}(t^{Y_n})\,\mathrm{d}t$.