Let $f$ be a function that expands as a power series on $D(0,R)$. Show that $\forall r \in [0, R[$, $|f(0)| \leqslant \sup_{t \in \mathbb{R}} |f(r\cos(t), r\sin(t))|$.

Throughout this part, $f$ denotes a function that expands in a power series on $D(0,R)$, i.e., $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ Show that $\forall r \in [0, R[$, $|f(0)| \leqslant \sup_{t \in \mathbb{R}} |f(r\cos(t), r\sin(t))|$.

Show an analogous result to $|f(0)| \leqslant \sup_{t \in \mathbb{R}} |f(r\cos(t), r\sin(t))|$ for harmonic functions.

Show an analogous result to Q34 for harmonic functions: for a harmonic function $g$ on $D(0,R)$, show that $\forall r \in [0, R[$, $|g(0)| \leqslant \sup_{t \in \mathbb{R}} |g(r\cos(t), r\sin(t))|$.

We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by $$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$ The purpose of Part III is to construct a function of class $C ^ { \infty }$ on $\mathbb { R }$, non-zero, whose all moments of order $p$ ($p \in \mathbb { N }$) are zero. Using the results of questions 36 and 37, conclude.

Let $f$ be defined on $I = ]-\pi/2, \pi/2[$ by $f(x) = \frac{\sin x + 1}{\cos x}$, and set $\alpha_n = f^{(n)}(0) = P_n(0)$ for every natural integer $n$. Using the identity $2f^{\prime}(x) = f(x)^2 + 1$, show $2\alpha_1 = \alpha_0^2 + 1$ and $$\forall n \in \mathbb{N}^{\star}, \quad 2\alpha_{n+1} = \sum_{k=0}^{n} \binom{n}{k} \alpha_k \alpha_{n-k}.$$

Let $\alpha_n = f^{(n)}(0)$ where $f(x) = \frac{\sin x + 1}{\cos x}$ on $I = ]-\pi/2, \pi/2[$. Let $R$ be the radius of convergence of the power series $\sum_{n \in \mathbb{N}} \frac{\alpha_n}{n!} x^n$ and $g$ its sum. Using Taylor's formula with integral remainder, show $$\forall N \in \mathbb{N}, \forall x \in \left[0, \pi/2\left[, \quad \sum_{n=0}^{N} \frac{\alpha_n}{n!} x^n \leqslant f(x)\right.\right.$$

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 $$\forall x \in I, \quad f(x) = g(x).$$

Using the fact that $f(x) = g(x)$ on $I = ]-\pi/2, \pi/2[$ where $f(x) = \frac{\sin x + 1}{\cos x}$ and $g$ is the sum of the power series $\sum_{n \in \mathbb{N}} \frac{\alpha_n}{n!} x^n$, deduce that $R = \pi/2$.

Using the decomposition of $f(x) = g(x) = \frac{\sin x + 1}{\cos x}$ into even and odd parts, deduce $$\forall x \in I, \quad \tan(x) = \sum_{n=0}^{+\infty} \frac{\alpha_{2n+1}}{(2n+1)!} x^{2n+1} \quad \text{and} \quad \frac{1}{\cos x} = \sum_{n=0}^{+\infty} \frac{\alpha_{2n}}{(2n)!} x^{2n}.$$

Let $t$ be the function defined on $I = ]-\pi/2, \pi/2[$ by $t(x) = \tan(x)$. For every natural integer $n$, express $t^{(n)}(0)$ as a function of the reals $(\alpha_i)_{i \in \mathbb{N}}$.

Recall, without justification, the expression of $t^{\prime}$ as a function of $t$, where $t(x) = \tan(x)$.

Using the expression of $t^{\prime}$ as a function of $t$ and the power series expansion $\tan(x) = \sum_{n=0}^{+\infty} \frac{\alpha_{2n+1}}{(2n+1)!} x^{2n+1}$, deduce $$\forall n \in \mathbb{N}^{\star}, \quad \alpha_{2n+1} = \sum_{k=1}^{n} \binom{2n}{2k-1} \alpha_{2k-1} \alpha_{2n-2k+1}.$$

2. a. Show that the radius of convergence of the power series $\sum \frac{E_n}{n!} x^n$ is $\geq 1$. b. For $|x| < 1$, we denote by $f(x)$ the sum of the preceding power series. Prove that $$2f'(x) = f^2(x) + 1, \quad \forall x \in ]-1, 1[$$ c. Deduce that $f(x) = \tan\left(\frac{x}{2} + \frac{\pi}{4}\right) = \frac{1}{\cos x} + \tan x, \quad \forall x \in ]-1, 1[$, then that $$\frac{1}{\cos x} = \sum_{n=0}^{\infty} \frac{E_{2n}}{(2n)!} x^{2n}, \quad \tan x = \sum_{n=0}^{\infty} \frac{E_{2n+1}}{(2n+1)!} x^{2n+1}, \quad \forall x \in ]-1, 1[$$

3. For a function $f : \mathbb{R} \rightarrow \mathbb{R}$ of class $C^\infty$ and $n \in \mathbb{N}$, we denote by $f^{(n)}$ the derivative of order $n$ of $f$, with the convention $f^{(0)} = f$. We denote by $D : \mathbb{R}[X] \rightarrow \mathbb{R}[X]$ the unique linear map such that $$D(X^0) = 0, \quad D(X^k) = k(X^{k-1} + X^{k+1}), \quad \forall k \in \mathbb{N}^*$$ For $n \in \mathbb{N}^*$, we denote by $D^n$ the composition of order $n$ of $D$, with the convention $D^0 = \mathrm{Id}$. a. Let $P_n = D^n(X)$. Prove that for $n \in \mathbb{N}$, $\tan^{(n)}(x) = P_n(\tan x)$ for $x \in ]-\frac{\pi}{2}, \frac{\pi}{2}[$. b. For $m \in \mathbb{N}^*$, let $V_m$ be the subspace of $\mathbb{R}[X]$ generated by $\{X, \ldots, X^m\}$. Let $\iota_m$ be the canonical injection of $V_m$ into $\mathbb{R}[X]$ and let $\tau_m : \mathbb{R}[X] \rightarrow V_m$ be the linear projection defined by $\tau_m(X^k) = X^k$ if $k \in \{1, \ldots, m\}$ and $\tau_m(X^k) = 0$ otherwise. Finally, we set $\delta_m = \tau_m \circ D \circ \iota_m$. Verify that $\delta_m$ is a linear map from $V_m$ to $V_m$ and write its matrix $M_m$ in the basis $(X, \ldots, X^m)$.

Show that for all $a \in \mathbb{R}$, $$\int_0^a \sin(x^2) \mathrm{d}x = \sum_{n=0}^{+\infty} (-1)^n \frac{a^{4n+3}}{(2n+1)!(4n+3)}$$

Show that for all $a \in \mathbb { R }$, $$\int _ { 0 } ^ { a } \sin \left( x ^ { 2 } \right) \mathrm { d } x = \sum _ { n = 0 } ^ { + \infty } ( - 1 ) ^ { n } \frac { a ^ { 4 n + 3 } } { ( 2 n + 1 ) ! ( 4 n + 3 ) }$$

Show that the limits $$\lim_{a \rightarrow +\infty} \int_0^a \sin(x^2) \mathrm{d}x \text{ and } \lim_{a \rightarrow +\infty} \int_0^a \cos(x^2) \mathrm{d}x$$ exist and are finite.

Show that the limits $$\lim _ { a \rightarrow + \infty } \int _ { 0 } ^ { a } \sin \left( x ^ { 2 } \right) \mathrm { d } x \quad \text { and } \lim _ { a \rightarrow + \infty } \int _ { 0 } ^ { a } \cos \left( x ^ { 2 } \right) \mathrm { d } x$$ exist and are finite.

From now on, $f$ denotes an infinitely differentiable function from $[0,1]$ to $\mathbb{R}$. We assume that there exists a unique point $x_0 \in [0,1]$ where $f'$ vanishes. We also assume that $f''(x_0) > 0$. We are also given an infinitely differentiable function $g : [0,1] \rightarrow \mathbb{R}$.
For all $x \in [x_0, 1]$, we define $$h(x) = \sqrt{|f(x) - f(x_0)|}$$ We admit that the bijection $h : [x_0, 1] \rightarrow [0, h(1)]$ admits an inverse application $h^{-1} : [0, h(1)] \rightarrow [x_0, 1]$ that is infinitely differentiable.
We assume that $x_0 \in ]0,1[$. Show that, as $t \rightarrow +\infty$, $$\int_0^1 g(x) \sin(tf(x)) \mathrm{d}x = g(x_0) \sin\left(tf(x_0) + \frac{\pi}{4}\right) \sqrt{\frac{2\pi}{tf''(x_0)}} + O\left(\frac{1}{t}\right)$$

From now on, $f$ denotes an infinitely differentiable function from $[ 0,1 ]$ to $\mathbb { R }$. We assume that there exists a unique point $x _ { 0 } \in \left[ 0,1 \left[ \right. \right.$ where $f ^ { \prime }$ vanishes. We also assume that $f ^ { \prime \prime } \left( x _ { 0 } \right) > 0$. We are also given an infinitely differentiable function $g : [ 0,1 ] \rightarrow \mathbb { R }$.
For all $x \in \left[ x _ { 0 } , 1 \right]$, we define $$h ( x ) = \sqrt { \left| f ( x ) - f \left( x _ { 0 } \right) \right| }$$ We admit that the bijection $$h : \left\{ \begin{array} { c c c } { \left[ x _ { 0 } , 1 \right] } & \rightarrow & { [ 0 , h ( 1 ) ] } \\ x & \mapsto & h ( x ) \end{array} \right.$$ admits an inverse map $h ^ { - 1 } : [ 0 , h ( 1 ) ] \rightarrow \left[ x _ { 0 } , 1 \right]$ that is infinitely differentiable.
Assume that $\left. x _ { 0 } \in \right] 0,1 [$. Show that, as $t \rightarrow + \infty$, $$\int _ { 0 } ^ { 1 } g ( x ) \sin ( t f ( x ) ) \mathrm { d } x = g \left( x _ { 0 } \right) \sin \left( t f \left( x _ { 0 } \right) + \frac { \pi } { 4 } \right) \sqrt { \frac { 2 \pi } { t f ^ { \prime \prime } \left( x _ { 0 } \right) } } + O \left( \frac { 1 } { t } \right)$$

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 $\mathbb { R }$.

We define a sequence $(a_n)_{n \geqslant 1}$ by setting $$\forall n \in \mathbb{N}^*, \quad a_n = \frac{(-n)^{n-1}}{n!}.$$ We define, when possible, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$, with radius of convergence $R$. Justify that the function $S$ is of class $\mathcal{C}^\infty$ on $]-R, R[$ and, for every integer $n \in \mathbb{N}$, express $S^{(n)}(0)$ as a function of $n$.

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 $f = \Gamma$.