We define for all real $x > 0$ the sequence $(I_{n}(x))_{n \geqslant 1}$ by: $$I_{n}(x) = \int_{0}^{n} \left(1 - \frac{t}{n}\right)^{n} t^{x-1} dt$$ Show that, for all $x > 0$, $$\lim_{n \rightarrow +\infty} I_{n}(x) = \Gamma(x)$$

We define for all real $x > 0$ the sequences $(I_{n}(x))_{n \geqslant 1}$ and $(J_{n}(x))_{n \geqslant 0}$ by: $$I_{n}(x) = \int_{0}^{n} \left(1 - \frac{t}{n}\right)^{n} t^{x-1} dt, \qquad J_{n}(x) = \int_{0}^{1} (1-t)^{n} t^{x-1} dt$$ Establish Euler's identity: $$\forall x > 0, \quad \Gamma(x) = \lim_{n \rightarrow +\infty} \frac{n! \, n^{x}}{x(x+1) \cdots (x+n)}$$

Let $h$ be a function of class $C^1$ on $\mathbb{R}^+$. We assume that $r \mapsto r^2 h(r)$ is bounded and we set $H(q) = \int_1^{+\infty} \frac{t\, h(qt)}{\sqrt{t^2-1}}\,\mathrm{d}t$.
Show that near $+\infty$ we have $H(q) = O\left(\frac{1}{q^2}\right)$.

Let $h$ be a function of class $C^1$ on $\mathbb{R}^+$. We assume that $r \mapsto r^2 h(r)$ is bounded and we set $H(q) = \int_1^{+\infty} \frac{t\, h(qt)}{\sqrt{t^2-1}}\,\mathrm{d}t$.
Prove that if we further assume that $r \mapsto r^4 h^{\prime}(r)$ is bounded, then the function $H$ is of class $C^1$ on $]0, +\infty[$.

We consider a function $f$ in $\mathcal{B}_1$ whose partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. We set, with the notations of part III: $$\forall q \in \mathbb{R}^+,\quad F(q) = \frac{1}{2\pi}\int_0^{2\pi} \hat{f}(q,\theta)\,\mathrm{d}\theta = 2\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r$$
Justify that $F$ is of class $C^1$ on $]0, +\infty[$ and that near $+\infty$ we have $F(q) = O\left(\frac{1}{q}\right)$.

We consider a function $f$ in $\mathcal{B}_1$ whose partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. We set: $$\forall q \in \mathbb{R}^+,\quad F(q) = \frac{1}{2\pi}\int_0^{2\pi} \hat{f}(q,\theta)\,\mathrm{d}\theta = 2\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r$$
Prove: $\forall \varepsilon > 0,\ \int_\varepsilon^{+\infty} \frac{F^{\prime}(q)}{q}\,\mathrm{d}q = -\frac{F(\varepsilon)}{\varepsilon} + 2\int_\varepsilon^{+\infty} \frac{1}{q^2}\left(\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r\right)\mathrm{d}q$.

We consider a function $f$ in $\mathcal{B}_1$ whose partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. We set: $$\forall q \in \mathbb{R}^+,\quad F(q) = \frac{1}{2\pi}\int_0^{2\pi} \hat{f}(q,\theta)\,\mathrm{d}\theta = 2\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r$$
We admit that we can interchange the two integrals and therefore that $$\forall \varepsilon > 0 \quad \int_\varepsilon^{+\infty} \left(\frac{1}{q^2}\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r\right)\mathrm{d}q = \int_\varepsilon^{+\infty} \left(\int_\varepsilon^r \frac{r\bar{f}(r)}{q^2\sqrt{r^2-q^2}}\,\mathrm{d}q\right)\mathrm{d}r$$
Deduce that $\forall \varepsilon > 0,\ \int_\varepsilon^{+\infty} \frac{F^{\prime}(q)}{q}\,\mathrm{d}q = -2\varepsilon \int_\varepsilon^{+\infty} \frac{\bar{f}(r)}{r\sqrt{r^2-\varepsilon^2}}\,\mathrm{d}r$.

We consider a function $f$ in $\mathcal{B}_1$ such that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. The Radon inversion formula states: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{-1}{\pi} \int_0^{+\infty} \frac{R_{x,y}^{\prime}(q)}{q}\,\mathrm{d}q$, where $R_{x,y}(q) = \frac{1}{2\pi}\int_0^{2\pi} \hat{f}(x\cos\theta + y\sin\theta + q, \theta)\,\mathrm{d}\theta$.
Establish the Radon inversion formula for this function $f$ at the point $(x,y) = (0,0)$.

We consider a function $f$ in $\mathcal{B}_1$ such that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. The Radon inversion formula states: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{-1}{\pi} \int_0^{+\infty} \frac{R_{x,y}^{\prime}(q)}{q}\,\mathrm{d}q$.
Are the hypotheses made on $f$ necessary for the Radon inversion formula to be verified at the point $(x,y) = (0,0)$?

We consider a function $f$ in $\mathcal{B}_1$ such that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. The Radon inversion formula states: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{-1}{\pi} \int_0^{+\infty} \frac{R_{x,y}^{\prime}(q)}{q}\,\mathrm{d}q$.
Propose a method to obtain the Radon inversion formula at any pair $(x,y)$ from the formula at $(0,0)$.

For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Show $$\left\{ \begin{array}{l} \forall x > 0, \quad f(x) = x \ln(x) - \frac{1}{2} x \ln\left(x^{2} + 1\right) - \arctan(x) + \frac{\pi}{2} \\ f(0) = \frac{\pi}{2} \end{array} \right.$$

For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Show $$\forall s \in \mathbb{R}, \quad |s| = \frac{2}{\pi} \int_{0}^{\infty} \frac{1 - \cos(st)}{t^{2}} \mathrm{~d}t$$

Prove that $\forall x \in \mathbb{R}, \quad \int_{-\infty}^{+\infty} \frac{e^{2\pi\mathrm{i} x\xi}}{1+(2\pi\xi)^{2}} \mathrm{d}\xi = \frac{1}{2} e^{-|x|}$.

In the general model of a Pólya urn ($b = c = 0$, $a = d$), the function $G$ is defined on $U$ by $$G(x,u,v) = u^{a_{0}} v^{b_{0}} (1 - axu^{a})^{-a_{0}/a} (1 - axv^{a})^{-b_{0}/a}$$ and admits the expansion $G(x,u,v) = \sum_{n=0}^{+\infty} Q_{n}(u,v) \frac{x^{n}}{n!}$ on $D_{\rho}$.
Prove that $G$ admits a first-order partial derivative with respect to $u$ on the domain $D_{\rho}$, obtained by term-by-term differentiation with respect to $u$ of the expression for $G$.

In the general model of a Pólya urn ($b = c = 0$, $a = d$), the function $G$ is defined on $U$ by $$G(x,u,v) = u^{a_{0}} v^{b_{0}} (1 - axu^{a})^{-a_{0}/a} (1 - axv^{a})^{-b_{0}/a}$$ and admits the expansion $G(x,u,v) = \sum_{n=0}^{+\infty} Q_{n}(u,v) \frac{x^{n}}{n!}$ on $D_{\rho}$. The function $H(x,u,v) = \sum_{n=0}^{+\infty} P_{n}(u,v) \frac{x^{n}}{n!}$ was defined in part III.
Deduce that, for all integers $n$, $P_{n} = Q_{n}$, and then that $H$ and $G$ coincide on $D_{\rho}$.

Show that for all $\theta \in ] - \pi ; \pi [$, the function $f$ defined by
$$\begin{aligned} f : ] 0 ; + \infty [ & \longrightarrow \mathbf { C } \\ t & \longmapsto \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \end{aligned}$$
is defined and integrable on $] 0 ; + \infty [$, where $x$ is a fixed element of $]0;1[$.

The solution for $x$ of the equation $\int _ { \sqrt { 2 } } ^ { x } \frac { d t } { t \sqrt { t ^ { 2 } - 1 } } = \frac { \pi } { 2 }$ is
(1) 2
(2) $\pi$
(3) $\frac { \sqrt { 3 } } { 2 }$
(4) None of these