The Euler Gamma function is defined, for all real $x > 0$, by: $$\Gamma(x) = \int_{0}^{+\infty} e^{-t} t^{x-1} dt$$ Justify that the function $\Gamma$ is of class $\mathcal{C}^{1}$ and strictly positive on $]0, +\infty[$.

We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
a) Let $m \in A$. Show that $T_{m}$ is continuous on $\mathbb{R}_{+}$.
b) Let $m \in \mathbb{R}$. Show that $T_{m}$ is continuous on $\mathbb{R}_{+}^{*}$.
c) Show that for $x \in \mathbb{R}_{+}^{*}$ and $m \in \mathbb{R}$, $$T_{m}(x) \geq e^{-1} \int_{0}^{1} t^{m} e^{-x/t} dt$$ Deduce the value of $\lim_{x \rightarrow 0^{+}} T_{m}(x)$ when $m \notin A$, using the change of variable $w = x/t$.

We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
a) Let $m \in \mathbb{R}$. Show that $T_{m}$ is of class $C^{1}$ on $\mathbb{R}_{+}^{*}$, and calculate $T_{m}^{\prime}$ in terms of $T_{m-1}$.
b) Let $m \in \mathbb{R}$. Is the function $T_{m}$ of class $C^{\infty}$ on $\mathbb{R}_{+}^{*}$? What is the monotonicity of $T_{m}$ on $\mathbb{R}_{+}^{*}$? Is the function $T_{m}$ convex on $\mathbb{R}_{+}^{*}$?
c) Discuss as a function of $m \in \mathbb{R}$ the right-differentiability of $T_{m}$ at 0.

We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
Let $L > 0$, $\rho \in C([0,L])$, and $g_{0}$ continuous and integrable on $\mathbb{R}_{+}$. We set for $x \in [0,L]$ and $v \in \mathbb{R}^{*}$: $$\begin{gathered} g(x,v) = \frac{2}{\sqrt{\pi}} \frac{e^{-v^{2}}}{|v|} \int_{x}^{L} \rho(y) e^{-\frac{x-y}{v}} dy, \quad \text{if} \quad v < 0, \\ g(x,v) = \frac{2}{\sqrt{\pi}} \frac{e^{-v^{2}}}{|v|} \int_{0}^{x} \rho(y) e^{-\frac{x-y}{v}} dy + g_{0}(v) e^{-\frac{x}{v}}, \quad \text{if} \quad v > 0. \end{gathered}$$
a) Show that $\alpha : x \in [0,L] \mapsto \int_{0}^{\infty} g_{0}(v) e^{-\frac{x}{v}} dv$ defines a function in $C([0,L])$.
b) Show that for $v \in \mathbb{R}^{*}$, the function $x \in [0,L] \mapsto g(x,v)$ is of class $C^{1}$ on $[0,L]$ and $$\begin{aligned} & \forall x \in [0,L], v \in \mathbb{R}^{*}, \quad v \frac{\partial g}{\partial x}(x,v) = \rho(x) \frac{2}{\sqrt{\pi}} e^{-v^{2}} - g(x,v) \\ & \forall v \in \mathbb{R}_{+}^{*}, \quad g(0,v) = g_{0}(v), \quad \forall v \in \mathbb{R}_{-}^{*}, \quad g(L,v) = 0 \end{aligned}$$

We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
We admit in this question the following theorem: Let $L > 0$. If $H$ is a mapping from $C([0,L])$ to $C([0,L])$ satisfying $$\exists k \in [0,1[, \quad \forall \rho_{1}, \rho_{2} \in C([0,L]), \quad \|H(\rho_{1}) - H(\rho_{2})\|_{\infty} \leq k \|\rho_{1} - \rho_{2}\|_{\infty}$$ then there exists a unique $\rho \in C([0,L])$ such that $H(\rho) = \rho$.
a) Let $L > 0$ and $\tilde{\alpha} \in C([0,L])$. Show that if $L \in ]0, 1/20[$, there exists a unique function $\tilde{\rho} \in C([0,L])$ such that $$\forall x \in [0,L], \quad \tilde{\rho}(x) = \tilde{\alpha}(x) + \frac{2}{\sqrt{\pi}} \int_{0}^{L} \tilde{\rho}(y) T_{-1}(|x-y|) dy$$
b) Let $L \in ]0, 1/20[$, and $g_{0}$ continuous and integrable on $\mathbb{R}_{+}$. Show that there exists a function $\tilde{g}$ from $[0,L] \times \mathbb{R}$ to $\mathbb{R}$ such that

• $\forall v \in \mathbb{R}^{*}$, $\tilde{g}(\cdot, v)$ is of class $C^{1}$ on $[0,L]$
• $\forall x \in [0,L]$, $\tilde{g}(x, \cdot)$ is integrable on $\mathbb{R}_{+}^{*}$ and $\mathbb{R}_{-}^{*}$
• $\forall x \in [0,L], v \in \mathbb{R}^{*}$, $v \frac{\partial \tilde{g}}{\partial x}(x,v) = \left(\int_{\mathbb{R}_{+}^{*}} \tilde{g}(x,w) dw + \int_{\mathbb{R}_{-}^{*}} \tilde{g}(x,w) dw\right) \frac{2}{\sqrt{\pi}} e^{-v^{2}} - \tilde{g}(x,v)$
• $\forall v \in \mathbb{R}_{+}^{*}$, $\tilde{g}(0,v) = g_{0}(v)$, $\quad \forall v \in \mathbb{R}_{-}^{*}$, $\tilde{g}(L,v) = 0$

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$. Throughout the rest of this question we assume $x > 1$ and $y > 1$. We denote $F _ { x , y }$ the antiderivative on $\mathbb { R } ^ { + }$ of $t \mapsto e ^ { - t } t ^ { x + y - 1 }$ which vanishes at 0.
Let $G ( a ) = \int _ { 0 } ^ { + \infty } \frac { u ^ { x - 1 } } { ( 1 + u ) ^ { x + y } } F _ { x , y } ( ( 1 + u ) a ) \mathrm { d } u$.
Show that $G$ is defined and continuous on $\mathbb { R } ^ { + }$.

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$. Throughout the rest of this question we assume $x > 1$ and $y > 1$. We denote $F _ { x , y }$ the antiderivative on $\mathbb { R } ^ { + }$ of $t \mapsto e ^ { - t } t ^ { x + y - 1 }$ which vanishes at 0, and $G ( a ) = \int _ { 0 } ^ { + \infty } \frac { u ^ { x - 1 } } { ( 1 + u ) ^ { x + y } } F _ { x , y } ( ( 1 + u ) a ) \mathrm { d } u$.
Show that $G$ is of class $\mathcal { C } ^ { 1 }$ on every segment $[ c , d ]$ included in $\mathbb { R } ^ { + * }$, then that $G$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R } ^ { + * }$.

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$. Throughout the rest of this question we assume $x > 1$ and $y > 1$. We denote $\Gamma ( x ) = \int _ { 0 } ^ { + \infty } t ^ { x - 1 } e ^ { - t } \mathrm {~d} t$, $F _ { x , y }$ the antiderivative on $\mathbb { R } ^ { + }$ of $t \mapsto e ^ { - t } t ^ { x + y - 1 }$ which vanishes at 0, and $G ( a ) = \int _ { 0 } ^ { + \infty } \frac { u ^ { x - 1 } } { ( 1 + u ) ^ { x + y } } F _ { x , y } ( ( 1 + u ) a ) \mathrm { d } u$.
Express for $a > 0$, $G ^ { \prime } ( a )$ as a function of $\Gamma ( x ) , e ^ { - a }$ and $a ^ { y - 1 }$.

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$. We have the relation $(\mathcal{R})$: $\beta ( x , y ) = \frac { \Gamma ( x ) \Gamma ( y ) } { \Gamma ( x + y ) }$. We define $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$.
From the relation $(\mathcal{R})$, justify that $\frac { \partial \beta } { \partial y }$ is defined on $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$.
Establish that for all real $x > 0$ and $y > 0 , \frac { \partial \beta } { \partial y } ( x , y ) = \beta ( x , y ) ( \psi ( y ) - \psi ( x + y ) )$.

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$.
Let $x > 0$ be fixed. What is the monotonicity on $\mathbb { R } ^ { + * }$ of the function $y \mapsto \beta ( x , y )$?

We define $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$ on $\mathbb{R}^{+*}$.
Show that the function $\psi$ is increasing on $\mathbb { R } ^ { + * }$.

We denote $B$ the function defined on $\mathbb { R } ^ { + * }$ by $B ( x ) = \frac { \partial ^ { 2 } \beta } { \partial y ^ { 2 } } ( x , 1 )$, where $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$ and $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$. We have $\frac { \partial \beta } { \partial y } ( x , y ) = \beta ( x , y ) ( \psi ( y ) - \psi ( x + y ) )$.
Justify that $B$ is defined on $\mathbb { R } ^ { + * }$.
Using the relation found in III.B.1, establish that for every real $x > 0$ $$x B ( x ) = ( \psi ( 1 + x ) - \psi ( 1 ) ) ^ { 2 } + \left( \psi ^ { \prime } ( 1 ) - \psi ^ { \prime } ( 1 + x ) \right)$$
Deduce that $B$ is $\mathcal { C } ^ { \infty }$ on $\mathbb { R } ^ { + * }$.

We denote $B$ the function defined on $\mathbb { R } ^ { + * }$ by $B ( x ) = \int _ { 0 } ^ { 1 } ( \ln ( 1 - t ) ) ^ { 2 } t ^ { x - 1 } \mathrm {~d} t$.
Give without justification an expression, using an integral, of $B ^ { ( p ) } ( x )$, for every natural integer $p$ and every real $x > 0$.

We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $$F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$$
Show that $F$ is well defined and of class $\mathscr { C } ^ { \infty }$ on $] 0 , + \infty [$.

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Show that $\Gamma$ is of class $\mathcal{C}^{\infty}$ on $\mathcal{D}$.
Let $k \in \mathbb{N}^{*}$ and $x \in \mathcal{D}$. Express $\Gamma^{(k)}(x)$, the $k$-th derivative of $\Gamma$ at point $x$, in the form of an integral.

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Show that $\Gamma^{\prime}$ vanishes at a unique real number $\xi$ whose integer part will be determined.

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 $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$, where $\mathrm{i}$ denotes the complex number with modulus 1 and argument $\pi/2$.
Show that the function $F : \begin{aligned} & \mathbb{R} \rightarrow \mathbb{C} \\ & x \mapsto F(x) \end{aligned}$ is defined and of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$.
Let $k$ be a non-zero natural number and let $x$ be a real number. Give an integral expression for $F^{(k)}(x)$, the $k$-th derivative of $F$ at $x$. Specify $F(0)$.

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 that $f$ is defined and continuous on $[0, +\infty[$ and of class $C^{2}$ on $]0, +\infty[$.

For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Express $f^{\prime\prime}$ on $]0, +\infty[$ using standard functions and deduce that $$\forall x > 0, \quad f^{\prime}(x) = \ln(x) - \frac{1}{2} \ln\left(x^{2} + 1\right)$$

Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
Justify that $f$ is of class $\mathcal{C}^2$, decreasing and convex on $I$.

Let $g$ be the function defined by
$$\begin{aligned} g : ] - \pi ; \pi [ & \longrightarrow \mathbf { C } \\ \theta & \longmapsto e ^ { \mathrm { i } x \theta } \int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \mathrm {~d} t \end{aligned}$$
where $x$ is a fixed element of $]0;1[$. Show that the function $g$ is of class $\mathcal { C } ^ { 1 }$ on $] - \pi ; \pi [$ and that for all $\theta \in ] - \pi ; \pi [$,
$$g ^ { \prime } ( \theta ) = \mathrm { i } e ^ { \mathrm { i } x \theta } \int _ { 0 } ^ { + \infty } h ^ { \prime } ( t ) \mathrm { d } t$$
where $h$ is the function defined by
$$\begin{aligned} h : ] 0 ; + \infty [ & \longrightarrow \mathbf { C } \\ t & \longmapsto \frac { t ^ { x } } { 1 + t e ^ { \mathrm { i } \theta } } . \end{aligned}$$
Calculate $h ( 0 )$ and
$$\lim _ { t \rightarrow + \infty } h ( t ) .$$
Deduce that the function $g$ is constant on $] - \pi ; \pi [$.

Let $f \in \mathcal { C } _ { b } ^ { 0 } ( [ 0 , + \infty [ )$. We define the Laplace transform of $f$ by the function $$\mathcal { L } ( f ) : t \in ] 0 , + \infty \left[ \mapsto \int _ { 0 } ^ { + \infty } e ^ { - t x } f ( x ) d x \right.$$ Prove that $\mathcal { L } ( f )$ is well-defined and of class $\mathcal { C } ^ { 1 }$ on $] 0 , + \infty [$, and express its derivative.

Problem 1: calculation of an integral
For $x \geqslant 0$ we define $$f ( x ) = \int _ { 0 } ^ { \infty } \frac { e ^ { - t x } } { 1 + t ^ { 2 } } \mathrm {~d} t \quad \text { and } \quad g ( x ) = \int _ { 0 } ^ { \infty } \frac { \sin t } { t + x } \mathrm {~d} t$$
Study of $f$. a. Show that $f ( x )$ is well defined for all $x \geqslant 0$. b. Show with precision that the function $f$ is of class $C ^ { 2 }$ on $] 0 , \infty [$, and also continuous at 0. c. Calculate $f + f ^ { \prime \prime }$ and deduce that $f$ is a solution of a linear second-order differential equation.