Show, for $r > 0$, that $$r < \rho(f) \Rightarrow \exists a > 0 \text{ such that } f \prec \frac{a}{r - z} \Rightarrow r \leqslant \rho(\hat{f})$$ deduce in particular that $\rho(\hat{f}) = \rho(f)$.

Show that $\widehat{f \cdot g} \prec \hat{f} \cdot \hat{g}$, deduce that $\rho(f \cdot g) \geqslant \min(\rho(f), \rho(g))$.

If $f \in O_n, n \geqslant 0, g \in O_1, h \in O_l, l \geqslant 1$ and $r \geqslant 1$, show that $h^r \in O_{rl}$, that $f \circ h \in O_{nl}$ and $f \circ (g + h) - f \circ g \in O_{n+l-1}$.

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Show that: $$\forall x \in ]-2\pi, 2\pi[ \backslash \{0\}, \quad \frac{x}{2} \operatorname{cotan}\left(\frac{x}{2}\right) = 1 - \sum_{k=1}^{+\infty} \frac{\zeta(2k)}{2^{2k-1}\pi^{2k}} x^{2k}.$$

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Deduce: $$\forall x \in ]-2\pi, 2\pi[ \backslash \{0\}, \quad \frac{ix}{e^{ix}-1} = 1 - \frac{ix}{2} - \sum_{k=1}^{+\infty} \frac{\zeta(2k)}{2^{2k-1}\pi^{2k}} \cdot x^{2k}.$$

Let $f$ and $g$ be power series, with $g \in O_1$. Show that $\widehat{f \circ g} \prec \hat{f} \circ \hat{g}$. Deduce that, if $f$ and $g$ have strictly positive radius of convergence, then $\rho(f \circ g) > 0$.

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$
Show that for all $z \in \mathbb{C}$ such that $|z| < 2\pi$, we have $$z = \left(e^z - 1\right)\left(1 - \frac{z}{2} + \sum_{k=1}^{+\infty} \frac{(-1)^{k-1} \zeta(2k)}{2^{2k-1}\pi^{2k}} z^{2k}\right).$$

If $f, g$ have non-negative real coefficients, $h, g \in O_1$, show that $h \prec g \Rightarrow f \circ h \prec f \circ g$.

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$
Show that the function $h$ is of class $C^\infty$ on $\mathbb{R}$ and that, for all $n \in \mathbb{N}^*$, we have $$h^{(2n)}(0) = \frac{(-1)^{n-1}(2n)!}{\pi^{2n} 2^{2n-1}} \zeta(2n).$$

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$
We define a sequence of real numbers $(b_n)_{n \in \mathbb{N}}$ by setting $b_0 = 1$, $b_1 = -\frac{1}{2}$, then $$\forall n \in \mathbb{N}^*, \quad b_{2n+1} = 0 \quad \text{and} \quad b_{2n} = \frac{(-1)^{n-1}(2n)! \zeta(2n)}{2^{2n-1}\pi^{2n}}.$$
Show that for all $n \in \mathbb{N}$: $$\sum_{k=0}^{n} \frac{b_k}{k!(n+1-k)!} = \begin{cases} 1 & \text{if } n = 0 \\ 0 & \text{if } n \geqslant 1 \end{cases}.$$

Show, if $f$ and $g \in O_1$ have non-negative real coefficients and if $r \in [0, \infty]$, that $f \circ g(r) = f(g(r))$.

Let $f$ and $g$ be power series, with $g \in O_1$. For all $z$ satisfying $|z| < \rho(\hat{f} \circ \hat{g})$, show that the series $f$ converges at $g(z)$ and that $f \circ g(z) = f(g(z))$.

Let $f, g$ and $h$ be power series, with $g, h \in O_1$, show that $(f \circ g) \circ h = f \circ (g \circ h)$.

We consider a power series $g \in O_1$ with non-negative real coefficients. We assume that there exist $a > 0, b > 0$ such that $$g \prec a\left(I + \frac{g^2}{b - g}\right)$$ Show that there exist $r > 0$ and a function $h: ]-r, r[ \longrightarrow \mathbb{R}$, expandable as a power series at 0, satisfying $h(0) = 0$ and such that $$h(x) = a\left(x + \frac{h(x)^2}{b - h(x)}\right)$$ for all $x \in ]-r, r[$. We also denote by $h$ the element of $O_1$ associated with the function $h$.

We consider a power series $g \in O_1$ with non-negative real coefficients. We assume that there exist $a > 0, b > 0$ such that $$g \prec a\left(I + \frac{g^2}{b - g}\right)$$ and $h$ is the function from question (11). Show, by induction on $k$, that $(g)_k \leqslant (h)_k$ for all $k \in \mathbb{N}$, conclude.

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 $t \in \left] - \pi / 2 , \pi / 2 \right[$, $$f ^ { ( k ) } ( \tan t ) = k ! \cos ^ { k + 1 } ( t ) \cos ( ( k + 1 ) t + k \pi / 2 ).$$

Let $a > 0$, $I = [-a, a]$, and $f(x) = \dfrac{1}{1+x^2}$ for $x \in \mathbb{R}$. For all $n \in \mathbb { N } ^ { * }$, let $P _ { n } = \Pi _ { n } ( f )$ be the Lagrange interpolation polynomial of $f$ on $I$. Show that, if $a < \frac { 1 } { 2 }$, the sequence of polynomials $\left( P _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ converges uniformly towards $f$ on $[ - a , a ]$.

We assume that $|\lambda| \notin \{0,1\}$ and that $\rho(f) > 0$. Using the result of question (22), conclude that the power series $h$ and $H$ of part E have strictly positive radius of convergence.

We now study the linearization problem in the case $|\lambda| = 1$, with $\lambda$ not a root of unity. We set, for $m \geqslant 1$, $$\alpha_m := \min(1/5, \omega_{m+1}, \omega_{m+2}, \ldots, \omega_{2m}), \quad \gamma_m := \alpha_m^{2/m},$$ where $\omega_k := |\lambda^k - \lambda|, k \geqslant 2$.
We are given a series $F \in O_{m+1}, m \geqslant 1$ such that $\rho(F) > 0$. Show that there exists $r_0 \in ]0,1[$ such that $\hat{F}(r) \leqslant r$ for all $r \in [0, r_0]$. Show then, for $\gamma \in ]0,1[$, that $$\hat{F}(r) \leqslant \gamma^m r$$ for all $r \in [0, \gamma r_0]$.

Let $\sum _ { k \geqslant 0 } c _ { k } x ^ { k }$ be a power series with radius of convergence $R > 0$ and $r \in ] 0 , R [$. Show that there exists $C \in \mathbb { R }$ such that $$\forall k \in \mathbb { N } , \quad \left| c _ { k } \right| \leqslant \frac { C } { r ^ { k } }.$$

We now study the linearization problem in the case $|\lambda| = 1$, with $\lambda$ not a root of unity. We set, for $m \geqslant 1$, $$\alpha_m := \min(1/5, \omega_{m+1}, \omega_{m+2}, \ldots, \omega_{2m}), \quad \gamma_m := \alpha_m^{2/m},$$ where $\omega_k := |\lambda^k - \lambda|, k \geqslant 2$, and $r_0 > 0$ is as given by question (24).
Still for $F \in O_{m+1}, m \geqslant 1$, we set $$P := \sum_{k=m+1}^{2m} \frac{(F)_k}{\lambda^k - \lambda} z^k \in O_{m+1} \quad , \quad R := (I + P)^\dagger - I.$$ Show that $P \circ (\lambda I) - \lambda P - F \in O_{2m+1}$ and that $R + P \in O_{2m+1}$. Show that $\hat{P}(r) \leqslant \alpha_m r$ for all $r \in [0, \gamma_m r_0]$, and that $$\hat{R}(r) \leqslant \frac{\alpha_m}{1 - \alpha_m} r$$ for all $r \in [0, (1-\alpha_m)\gamma_m r_0]$.

Let $\sum _ { k \geqslant 0 } c _ { k } x ^ { k }$ be a power series with radius of convergence $R > 0$, $r \in ]0, R[$, and $f(x) = \sum_{k=0}^{+\infty} c_k x^k$ for $x \in ]-R, R[$. Let $C \in \mathbb{R}$ be such that $|c_k| \leq C/r^k$ for all $k \in \mathbb{N}$. Deduce that for all $x \in ] - r , r [$ and for all $n \in \mathbb { N }$, $$\left| f ^ { ( n ) } ( x ) \right| \leqslant \frac { n ! r C } { ( r - | x | ) ^ { n + 1 } }.$$

We now study the linearization problem in the case $|\lambda| = 1$, with $\lambda$ not a root of unity. We set, for $m \geqslant 1$, $$\alpha_m := \min(1/5, \omega_{m+1}, \omega_{m+2}, \ldots, \omega_{2m}), \quad \gamma_m := \alpha_m^{2/m},$$ where $\omega_k := |\lambda^k - \lambda|, k \geqslant 2$, and $P, R$ as defined in question (25).
For $F \in O_{m+1}, m \geqslant 1$, show that $$G := (I + P)^\dagger \circ (\lambda I + F) \circ (I + P) - \lambda I = (I + R) \circ (\lambda I + F) \circ (I + P) - \lambda I$$ satisfies $G \in O_{2m+1}$.

Let $\sum _ { k \geqslant 0 } c _ { k } x ^ { k }$ be a power series with radius of convergence $R > 0$, $f(x) = \sum_{k=0}^{+\infty} c_k x^k$ for $x \in ]-R, R[$, and $a > 0$. Assume that $a < R / 3$. Show that the sequence of polynomials $\left( P _ { n } \right) _ { n \in \mathbb { N } ^ { * } } = \left( \Pi _ { n } ( f ) \right) _ { n \in \mathbb { N } ^ { * } }$ converges uniformly towards $f$ on $[ - a , a ]$.

We now study the linearization problem in the case $|\lambda| = 1$, with $\lambda$ not a root of unity. We set, for $m \geqslant 1$, $$\alpha_m := \min(1/5, \omega_{m+1}, \omega_{m+2}, \ldots, \omega_{2m}), \quad \gamma_m := \alpha_m^{2/m},$$ where $\omega_k := |\lambda^k - \lambda|, k \geqslant 2$, and $G$ as defined in question (26).
Show that $$\hat{G}(r) \leqslant \left(\alpha_m + (1 + \alpha_m)\alpha_m^2 + \frac{\alpha_m(1 + \alpha_m)(1 + \alpha_m^2)}{1 - \alpha_m}\right) r \leqslant r$$ for all $r$ such that $$0 \leqslant r \leqslant \frac{1 - \alpha_m}{(1 + \alpha_m)(1 + \alpha_m^2)} \gamma_m r_0$$