Let $f(x) = \frac{\sin x + 1}{\cos x}$ on $I = ]-\pi/2, \pi/2[$ and $g$ the sum of the power series $\sum_{n \in \mathbb{N}} \frac{\alpha_n}{n!} x^n$. Both satisfy $2h^{\prime}(x) = h(x)^2 + 1$. By considering the functions $\arctan f$ and $\arctan g$, show $$\forall x \in I, \quad f(x) = g(x).$$

If $a$ and $b$ are two real numbers, we denote $K _ { a , b }$ the function defined for all real $t$ by $K _ { a , b } ( t ) = \begin{cases} \frac { \mathrm { e } ^ { \mathrm { i } t b } - \mathrm { e } ^ { \mathrm { i } t a } } { 2 \mathrm { i } t } & \text { if } t \neq 0 , \\ \frac { b - a } { 2 } & \text { if } t = 0 . \end{cases}$ Using power series, show that $K _ { a , b }$ is of class $C ^ { \infty }$ on $\mathbb { R }$.

Let $\Gamma$ be the pointwise limit on $]0, +\infty[$ of the sequence $\left(\Gamma_n\right)_{n \geqslant 1}$ where $$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$ Let $f : ]0, +\infty[ \rightarrow ]0, +\infty[$ be a function of class $\mathscr{C}^2$ such that the function $\ln(f)$ is convex and satisfies $f(1) = 1$ and $f(x+1) = xf(x)$ for all $x > 0$.
Show that the function $$S : \begin{array}{ccc} ]0, +\infty[ & \longrightarrow & \mathbb{R} \\ x & \longmapsto & \ln\left(\frac{f(x)}{\Gamma(x)}\right) \end{array}$$ is 1-periodic and convex.

Let $\Gamma$ be the pointwise limit on $]0, +\infty[$ of the sequence $\left(\Gamma_n\right)_{n \geqslant 1}$ where $$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$ Let $f : ]0, +\infty[ \rightarrow ]0, +\infty[$ be a function of class $\mathscr{C}^2$ such that the function $\ln(f)$ is convex and satisfies $f(1) = 1$ and $f(x+1) = xf(x)$ for all $x > 0$. The function $S(x) = \ln\left(\frac{f(x)}{\Gamma(x)}\right)$ is 1-periodic and convex.
Deduce that $f = \Gamma$.

Using the result that for $x \in ]-\frac{1}{2}, \frac{1}{2}[$: $$\frac{\pi}{\cos(\pi x)} = \sum_{k=0}^{+\infty} \left(\sum_{n=0}^{+\infty} \frac{(-1)^n}{(2n+1)^{2k+1}}\right) 2^{2k+2} x^{2k},$$ deduce that the function $$v : \begin{array}{ccc} ]-\frac{\pi}{2}, \frac{\pi}{2}[ & \longrightarrow & \mathbb{R} \\ x & \longmapsto & \frac{1}{\cos(x)} \end{array}$$ is expandable as a power series and that, for all $k \in \mathbb{N}$, $$\sum_{n=0}^{+\infty} \frac{(-1)^n}{(2n+1)^{2k+1}} = \frac{\pi^{2k+1}}{2^{2k+2}(2k)!} E_{2k}$$ where, for all $k \in \mathbb{N}$, $E_{2k} = v^{(2k)}(0)$.

Let $a > 0$, $I = [-a, a]$, and $$f : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ x & \mapsto & \dfrac{1}{1+x^2} \end{array}.$$ Show that $f$ is of class $\mathcal { C } ^ { \infty }$ and that, for all $k$ in $\mathbb { N }$ and all $t \in \left] - \pi / 2 , \pi / 2 \right[$, $$f ^ { ( k ) } ( \tan t ) = k ! \cos ^ { k + 1 } ( t ) \cos ( ( k + 1 ) t + k \pi / 2 ).$$

For $p \in \mathbb { N } ^ { * }$, consider a polynomial $P \in \mathbb { R } [ X ]$ such that the polynomial function $x \mapsto P ( x )$ is a solution of the equation $\left( E _ { p } \right) : x \left( y ^ { \prime \prime } - y ^ { \prime } \right) + p y = 0$. For all $x \in \mathbb { R }$, we denote $h ( x ) = \mathrm { e } ^ { - x } P ( x )$. Justify that the function $h$ is developable as a power series on $\mathbb { R }$.

For $p \in \mathbb { N } ^ { * }$, let $h(x) = \mathrm{e}^{-x} P(x)$ where $P$ is a polynomial solution of $(E_p)$, with power series coefficients satisfying $b _ { n } = \frac { ( - 1 ) ^ { n - 1 } ( n + p - 1 ) ! } { p ! n ! ( n - 1 ) ! } b _ { 1 }$ for all $n \in \mathbb{N}^*$. We set $g _ { p } ( x ) = x ^ { p - 1 } \mathrm { e } ^ { - x }$. Justify that $g _ { p } ^ { ( p ) }$ is developable as a power series and deduce from Question 34 that, for all $x \in \mathbb { R }$, $$P ( x ) = C x \mathrm { e } ^ { x } g _ { p } ^ { ( p ) } ( x )$$ where $C$ is a real constant whose expression in terms of $b _ { 1 }$ and $p$ we will specify.

Let $t \in \mathbf{R}$ and $(i,j) \in \llbracket 1;N \rrbracket^2$, justify that the series $\sum_{n \geq 0} \frac{t^n K^n[i,j]}{n!}$ converges. We denote by $H_t \in \mathscr{M}_N(\mathbf{R})$ the matrix defined by $$\forall (i,j) \in \llbracket 1;N \rrbracket^2, H_t[i,j] = e^{-t} \sum_{n=0}^{+\infty} \frac{t^n K^n[i,j]}{n!}$$

Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
Prove that $f$ is expandable as a power series on $]-1, 1[$.

Let $\varphi_{0}$ be the function defined on $\mathbb{R}$ by
$$\left\{ \begin{array}{l} \varphi_{0}(x) = e^{-1/x^{2}} \text{ if } x \neq 0 \\ \varphi_{0}(0) = 0 \end{array} \right.$$
a. Show that for all $n \in \mathbb{N}$ there exists a polynomial $P_{n}$ such that for $x \neq 0$ we have
$$\varphi_{0}^{(n)}(x) = P_{n}\left(\frac{1}{x}\right) e^{-1/x^{2}}$$
b. Show that $\varphi_{0}$ is of class $C^{\infty}$ on $\mathbb{R}$.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function of class $C^{\infty}$ with compact support. For all $n \in \mathbb{N}$, we set $$M_{n} = \sup_{x \in \mathbb{R}} \left|f^{(n)}(x)\right| = \left\|f^{(n)}\right\|_{\infty}$$ In this part we assume that $f$ is not identically zero.
Deduce that for all $B > 0$ we have
$$\frac{M_{n}}{B^{n} n!} \underset{n \rightarrow \infty}{\longrightarrow} +\infty$$

We denote $\omega(t) = e^{2i\pi t}$ for $t \in [0,1]$. For all $p \in \mathbb{Z}$, we set $$I_{p} = \int_{0}^{1} \frac{\omega(t)^{p+1}}{\mathrm{e}^{\omega(t)} - 1} \,\mathrm{d}t.$$ Verify that this integral is well defined for all $p \in \mathbb{Z}$.

Let $\psi$ be the function from $\mathbb{R}$ to $\mathbb{R}$ such that, for all $x \in \mathbb{R}$, $$\psi(x) = \begin{cases} \dfrac{x}{\mathrm{e}^{x} - 1} & \text{if } x \neq 0 \\ 1 & \text{otherwise} \end{cases}$$ Let furthermore $u$ be the function from $\mathbb{R}^{2}$ to $\mathbb{R}$ such that, for all $(x,t) \in \mathbb{R}^{2}$, $$u(x,t) = \psi(x)\,\mathrm{e}^{tx}.$$ Show that $u$ is of class $\mathcal{C}^{\infty}$ on $\mathbb{R}^{2}$.

Let $f \in C^{2}(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f^{\prime}$ and $f^{\prime\prime}$ have slow growth and $t \in \mathbf{R}_{+}$.
Show that $x \in \mathbb{R} \mapsto P_{t}(f)(x)$ is of class $C^{2}$ on $\mathbf{R}$. Also show that
$$\forall x \in \mathbf{R}, \quad P_{t}(f)^{\prime}(x) = \mathrm{e}^{-t} \int_{-\infty}^{+\infty} f^{\prime}\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y$$
and
$$\forall x \in \mathbf{R}, \quad P_{t}(f)^{\prime\prime}(x) = \mathrm{e}^{-2t} \int_{-\infty}^{+\infty} f^{\prime\prime}\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y.$$

Let $f \in C^2(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f'$ and $f''$ have slow growth and $t \in \mathbf{R}_+$. Show that $x \in \mathbf{R} \mapsto P_t(f)(x)$ is of class $C^2$ on $\mathbf{R}$. Also show that $$\forall x \in \mathbf{R}, \quad P_t(f)'(x) = \mathrm{e}^{-t}\int_{-\infty}^{+\infty} f'\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y$$ and $$\forall x \in \mathbf{R}, \quad P_t(f)''(x) = \mathrm{e}^{-2t}\int_{-\infty}^{+\infty} f''\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y.$$

Show that $v(x)$ is not the power series expansion of a rational fraction.