We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ We define the sequence of polynomials $\left(L_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} L_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad L_n = \frac{1}{P_n^{(n)}(1)} P_n^{(n)} \end{array}\right.$$
II.D.1) For all $n \in \mathbb{N}$, we set $I_n = \int_0^1 P_n(u) \, du$.
Calculate, for all $n \in \mathbb{N}$, the value of $I_n$.
II.D.2) Deduce for all $n \in \mathbb{N}$ the relation: $\langle L_n, L_n \rangle = \frac{1}{2n+1}$.

Let $k \in \mathbb { N } ^ { * }$. We denote by $c _ { k }$ the positive real number such that: $c _ { k } \int _ { 0 } ^ { 2 \pi } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k } d t = 1$, and for every real $t$ we set $R _ { k } ( t ) = c _ { k } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k }$.
Calculate $\int _ { 0 } ^ { \pi } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k } \sin t \, d t$. Deduce that $c _ { k } \leq \frac { k + 1 } { 4 }$.

Study the convergence of the improper integrals $$\int _ { 0 } ^ { \pi } \ln ( \sin \theta ) d \theta \quad \int _ { 0 } ^ { \pi } \ln ( 1 - \cos \theta ) d \theta \quad \int _ { 0 } ^ { \pi } \ln ( 1 + \cos \theta ) d \theta$$
Deduce that, for all $x \in \mathbb { R }$, the integral $\int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$ converges.

Show that, as $x$ tends to $+ \infty$, $$2 \pi \ln ( x ) - \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$$ has a limit, which one will determine.

Deduce that, for all $x \in ] - 1,1 [$, we have $\int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta = 0$.
Deduce the value of $\int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$ in the case $| x | > 1$.

Show that the improper integral $\int _ { 0 } ^ { \pi / 2 } \ln ( \cos \theta ) d \theta$ converges.

Show that $\int _ { 0 } ^ { \pi } \ln ( \sin \theta ) \mathrm { d } \theta = 2 \int _ { 0 } ^ { \pi / 2 } \ln ( \sin \theta ) \mathrm { d } \theta = 2 \int _ { 0 } ^ { \pi / 2 } \ln ( \cos \theta ) \mathrm { d } \theta$.

Deduce that $\int _ { 0 } ^ { \pi } \ln ( \sin \theta ) \mathrm { d } \theta = - \pi \ln 2$.

Deduce that $\int _ { 0 } ^ { \pi } \ln ( 2 - 2 \cos \theta ) \mathrm { d } \theta = \int _ { 0 } ^ { \pi } \ln ( 2 + 2 \cos \theta ) \mathrm { d } \theta = 0$.

Let $f$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$.
Deduce that $\forall x \in \mathbb { R } \backslash \{ - 1,1 \}$ $$f ^ { \prime } ( x ) = 4 \int _ { 0 } ^ { + \infty } \frac { ( x + 1 ) t ^ { 2 } + ( x - 1 ) } { \left( ( x + 1 ) ^ { 2 } t ^ { 2 } + ( x - 1 ) ^ { 2 } \right) \left( t ^ { 2 } + 1 \right) } \mathrm { d } t$$

Let $f$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$.
Deduce that $$f ( x ) = \begin{cases} 2 \pi \ln ( | x | ) & \text { if } | x | > 1 \\ 0 & \text { if } | x | < 1 \end{cases}$$
One will first determine coefficients $A$ and $B$ as functions of $x$ such that $\frac { ( x + 1 ) T + ( x - 1 ) } { \left( ( x + 1 ) ^ { 2 } T + ( x - 1 ) ^ { 2 } \right) ( T + 1 ) } = \frac { A } { ( x + 1 ) ^ { 2 } T + ( x - 1 ) ^ { 2 } } + \frac { B } { T + 1 }$ for all $T \in \mathbb { R }$ such that these fractions are defined.

Let $f$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$.
Show that $f$ is continuous on $\mathbb { R }$ and that $f ( 1 ) = f ( - 1 ) = 0$.
One may show that $\forall x \in \mathbb { R } , x ^ { 2 } - 2 x \cos \theta + 1 \geqslant \sin ^ { 2 } \theta$ and use the dominated convergence theorem.

Show that $\forall x \in \mathbb { R } \backslash \{ - 1,1 \}$ $$\int _ { 0 } ^ { 2 \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta = \lim _ { n \rightarrow + \infty } \left( \frac { 2 \pi } { n } \sum _ { k = 1 } ^ { n } \ln \left( x ^ { 2 } - 2 x \cos \frac { 2 k \pi } { n } + 1 \right) \right)$$

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 may freely use the formula $T_{0}(0) = \frac{\sqrt{\pi}}{2}$.
a) Show that if $x \in \mathbb{R}_{+}^{*}$ and $m \in \mathbb{R}$, the integral defining $T_{m}(x)$ is convergent.
b) What is the interval $A$ of $m \in \mathbb{R}$ such that the integral defining $T_{m}(0)$ is convergent?
c) Calculate $T_{2k}(0)$ and $T_{2k+1}(0)$ for $k \in \mathbb{N}$ (in terms of $k!$ and $(2k)!$).

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 $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
a) Let $x \in \mathbb{R}_{+}^{*}$ and $m \in \mathbb{R}$. Calculate $T_{m}(x)$ in terms of $T_{m-2}(x)$ and $T_{m-3}(x)$. For this, you may consider the quantity $$\int_{A}^{B} t^{m-1}\left(2t - x/t^{2}\right) e^{-\left(t^{2}+x/t\right)} dt$$ for $0 < A < B$.
b) Let $m \in \mathbb{R}$. Find a relation between $x T_{m}^{\prime\prime\prime}(x)$, $T_{m}^{\prime\prime}(x)$ and $T_{m}(x)$ for $x \in \mathbb{R}_{+}^{*}$.

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 $x \in \mathbb{R}_{+}^{*}$ and $m \in \mathbb{R}$. Perform the change of variable $t = 1/u$ in the integral defining $T_{m}$. Justify the calculation carefully.
b) Let $n \in \mathbb{N}$. Justify the existence of the quantity $\int_{0}^{\infty} u^{n} e^{-u} du$ and calculate it.
c) Show that for $m \in \mathbb{N} - \{0,1\}$, $T_{-m}(1) \leq (m-2)!$.
d) Let $k \in \mathbb{N}$. Show that the radius of convergence $R$ of the power series $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!} T_{k-n}(1) x^{n}$ satisfies $R \geq 1$.
e) Let $k \in \mathbb{N}$. Show that for $x \in ]-1,1[$, $$T_{k}(1+x) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!} T_{k-n}(1) x^{n}$$

We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$ For $t, x \in \mathbb{R}_{+}^{*}$, we set $g(t,x) = t^{2} + x/t$.
a) Show that for $x \in \mathbb{R}_{+}^{*}$, the function $t \in \mathbb{R}_{+}^{*} \mapsto g(t,x)$ is convex and admits a unique minimum at $t = M(x)$, which we will determine. Calculate $g(M(x),x)$.
b) Let $m \in \mathbb{R}_{+}$, and $x \in \mathbb{R}_{+}^{*}$. Show using the inequality $M(x) \leq x^{1/3}$ that $$T_{m}(x) \leq \int_{0}^{x^{1/3}} t^{m} e^{-g(t,x)} dt + e^{-\frac{19}{10} x^{2/3}} \left(\int_{x^{1/3}}^{\infty} t^{m} e^{-\frac{t^{2}}{20}} dt\right)$$
c) Show that for every $\varepsilon > 0$ (and $m \in \mathbb{R}_{+}$), we can find $C > 0$ such that $$\forall t \geq 1, \quad t^{m} \leq C t e^{\varepsilon t^{2}}$$ Deduce that for every $\varepsilon > 0$, $$e^{-\frac{19}{10} x^{2/3}} \left(\int_{x^{1/3}}^{\infty} t^{m} e^{-\frac{t^{2}}{20}} dt\right) = O_{x \rightarrow \infty}\left(e^{-\left[\frac{39}{20} - \varepsilon\right] x^{2/3}}\right)$$
d) Show that $$T_{m}(x) = O_{x \rightarrow \infty}\left(x^{\frac{m+1}{3}} e^{-3(x/2)^{2/3}}\right)$$

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

Justify the existence of the integral $\int_1^{+\infty} \frac{\mathrm{d}t}{t\sqrt{t^2-1}}$ and show that its value is $\frac{\pi}{2}$.

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.

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 $] 0 , + \infty [$.

We study the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by $$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$$ Show that $u_{1} = u_{2} = \frac{\pi}{2}$.