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Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2024 centrale-maths2__official

29 maths questions

Q1 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Show that $\Delta$ is an endomorphism of $\mathbb{K}[X]$, where $$\Delta : \begin{cases} \mathbb{K}[X] \rightarrow \mathbb{K}[X] \\ P(X) \mapsto P(X+1) - P(X) \end{cases}$$
Q2 Factor & Remainder Theorem Polynomial Degree and Structural Properties View
Let $P \in \mathbb{K}[X]$. Determine the degree of $\Delta(P)$ as a function of that of $P$, where $\Delta(P) = P(X+1) - P(X)$.
Q3 Factor & Remainder Theorem Polynomial Degree and Structural Properties View
Show that, for all $d \in \mathbb{N}^{*}$, $\Delta$ induces an endomorphism on $\mathbb{K}_{d}[X]$, where $\Delta(P) = P(X+1) - P(X)$.
Q4 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
We denote by $\Delta_{d}$ the endomorphism of $\mathbb{K}_{d}[X]$ induced by $\Delta$, where $\Delta(P) = P(X+1) - P(X)$. Determine $\operatorname{Ker}(\Delta_{d})$ and $\operatorname{Im}(\Delta_{d})$ for all $d \in \mathbb{N}^{*}$.
Q5 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
We consider the application $\Delta$ defined by $\Delta(P) = P(X+1) - P(X)$. Deduce $\operatorname{Ker}(\Delta)$ and $\operatorname{Im}(\Delta)$. Apply the results obtained to the study of the equation $(E_{h})$: $$\forall x \in \mathbb{K},\, f(x+1) - f(x) = h(x)$$ in the case where $h$ is a polynomial function.
Q6 Sequences and Series Evaluation of a Finite or Infinite Sum View
Suppose (for this question only) that $h$ is the function $x \mapsto x$. Determine a solution of $(E_{h})$: $$\forall x \in \mathbb{K},\, f(x+1) - f(x) = x$$ in $\mathbb{K}_{2}[X]$, then all polynomial solutions of the equation $(E_{h})$.
Q7 Factor & Remainder Theorem Proof of Polynomial Divisibility or Identity View
We denote by $\Delta_{d}$ the endomorphism of $\mathbb{K}_{d}[X]$ induced by $\Delta$, where $\Delta(P) = P(X+1) - P(X)$. Let $d \in \mathbb{N}^{*}$. Determine an annihilating polynomial of $\Delta_{d}$. Is the endomorphism $\Delta_{d}$ diagonalisable?
Q8 Exponential Functions Algebraic Simplification and Expression Manipulation View
We denote by $\mathcal{E}$ the set of functions $f : \mathbb{C} \rightarrow \mathbb{C}$ expandable as a power series with radius of convergence infinity. Justify that if $(f, g) \in \mathcal{E}^{2}$ and $(\lambda, \mu) \in \mathbb{C}^{2}$, then $\lambda f + \mu g \in \mathcal{E}$ and $fg \in \mathcal{E}$.
Q9 Taylor series Derive series via differentiation or integration of a known series View
We denote by $\mathcal{E}$ the set of functions $f : \mathbb{C} \rightarrow \mathbb{C}$ expandable as a power series with radius of convergence infinity, and $\omega(t) = e^{2i\pi t}$ for $t \in [0,1]$. Let $f \in \mathcal{E}$ whose power series expansion we denote $\sum a_{n} z^{n}$. Show that, for all $k \in \mathbb{Z}$: $$\int_{0}^{1} f(\omega(t)) \omega(t)^{-k} \,\mathrm{d}t = \begin{cases} a_{k} & \text{if } k \in \mathbb{N} \\ 0 & \text{otherwise} \end{cases}$$
Q10 Taylor series Prove smoothness or power series expandability of a function View
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 $p \in \mathbb{Z}$.
Q11 Complex Numbers Arithmetic Existence Theorems and Advanced Proof (e.g., Fundamental Theorem of Algebra) View
We denote $\mathbb{U}$ the multiplicative group of complex numbers of modulus 1. Show that there exist a function $\beta \in \mathcal{E}$ and a constant $C \in ]0,1[$ such that, for all $\zeta \in \mathbb{U}$, $$\mathrm{e}^{\zeta} - 1 = \zeta(1 + \zeta \beta(\zeta)) \quad \text{and} \quad |\beta(\zeta)| \leqslant C.$$
Q12 Taylor series Construct series for a composite or related function View
Using the result of Q11, deduce that for all $\zeta \in \mathbb{U}$ and all $p \in \mathbb{Z}$, $$\frac{\zeta^{p}}{\mathrm{e}^{\zeta} - 1} = \sum_{j=0}^{+\infty} (-1)^{j} \zeta^{j+p-1} \beta(\zeta)^{j}$$ where $\beta \in \mathcal{E}$ and $|\beta(\zeta)| \leqslant C < 1$ for all $\zeta \in \mathbb{U}$.
Q13 Taylor series Extract derivative values from a given series View
We denote $\omega(t) = e^{2i\pi t}$ for $t \in [0,1]$, and for all $p \in \mathbb{Z}$, $$I_{p} = \int_{0}^{1} \frac{\omega(t)^{p+1}}{\mathrm{e}^{\omega(t)} - 1} \,\mathrm{d}t.$$ Show that $I_{0} = 1$ and that, for all $p \in \mathbb{N}^{*}$, $I_{p} = 0$.
Q14 Sequences and Series Power Series Expansion and Radius of Convergence View
For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, we define $$B_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\omega(t)}}{(\mathrm{e}^{\omega(t)} - 1)\omega(t)^{n-1}} \,\mathrm{d}t$$ where $\omega(t) = e^{2i\pi t}$, and for all $p \in \mathbb{Z}$, $$I_{p} = \int_{0}^{1} \frac{\omega(t)^{p+1}}{\mathrm{e}^{\omega(t)} - 1} \,\mathrm{d}t.$$ Show that, for all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, $$B_{n}(z) = n! \sum_{k=0}^{n} \frac{z^{k}}{k!} I_{k-n}.$$ Deduce that $B_{n}$ is a monic polynomial of degree $n$.
Q15 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, the Bernoulli polynomial is defined by $$B_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\omega(t)}}{(\mathrm{e}^{\omega(t)} - 1)\omega(t)^{n-1}} \,\mathrm{d}t$$ where $\omega(t) = e^{2i\pi t}$. Show that, for all $n \in \mathbb{N}^{*}$, $B_{n}' = n B_{n-1}$.
Q16 Taylor series Formal power series manipulation (Cauchy product, algebraic identities) View
For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, the Bernoulli polynomial is defined by $$B_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\omega(t)}}{(\mathrm{e}^{\omega(t)} - 1)\omega(t)^{n-1}} \,\mathrm{d}t$$ where $\omega(t) = e^{2i\pi t}$. Show that, for all $n \in \mathbb{N}^{*}$ and all $z \in \mathbb{C}$, $$B_{n}(z+1) - B_{n}(z) = n z^{n-1}.$$
Q17 Sequences and Series Evaluation of a Finite or Infinite Sum View
Using the Bernoulli polynomials $(B_n)_{n \in \mathbb{N}}$ satisfying $B_n(z+1) - B_n(z) = nz^{n-1}$ for all $n \in \mathbb{N}^*$, deduce the expression of a polynomial function satisfying the equation $(E_h)$: $$\forall x \in \mathbb{C},\, f(x+1) - f(x) = h(x)$$ on $\mathbb{C}$ when $h$ is a polynomial function.
Q18 Sequences and Series Recurrence Relations and Sequence Properties View
Show that $(B_{n})_{n \in \mathbb{N}}$ is the unique sequence of polynomials satisfying $$\begin{cases} B_{0} = 1 \\ \forall n \in \mathbb{N}^{*},\, B_{n}' = n B_{n-1} \\ \forall n \in \mathbb{N}^{*},\, \displaystyle\int_{0}^{1} B_{n}(t)\,\mathrm{d}t = 0 \end{cases}$$
Q19 Sequences and Series Functional Equations and Identities via Series View
Let $(H_{n})_{n \in \mathbb{N}}$ be the sequence of polynomials defined by: $\forall n \in \mathbb{N},\, H_{n}(X) = (-1)^{n} B_{n}(1-X)$, where $(B_n)_{n\in\mathbb{N}}$ are the Bernoulli polynomials satisfying $$\begin{cases} B_{0} = 1 \\ \forall n \in \mathbb{N}^{*},\, B_{n}' = n B_{n-1} \\ \forall n \in \mathbb{N}^{*},\, \displaystyle\int_{0}^{1} B_{n}(t)\,\mathrm{d}t = 0 \end{cases}$$ Show that for all $n \in \mathbb{N}$, $H_{n} = B_{n}$.
Q20 Taylor series Prove smoothness or power series expandability of a function View
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 $\mathbb{R}^{2}$.
Q21 Taylor series Recursive or implicit derivative computation for series coefficients View
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 $(x,t) \in \mathbb{R}^{2}$, $$u(x,t) = \psi(x)\,\mathrm{e}^{tx}.$$ For all $(x,t) \in \mathbb{R}^{2}$, calculate $\dfrac{\partial u}{\partial t}(x,t)$ then show that, for all $n \in \mathbb{N}^{*}$, $$\frac{\partial}{\partial t}\frac{\partial^{n} u}{\partial x^{n}}(x,t) = x\frac{\partial^{n} u}{\partial x^{n}}(x,t) + n\frac{\partial^{n-1} u}{\partial x^{n-1}}(x,t).$$
Q22 Taylor series Recursive or implicit derivative computation for series coefficients View
Let $\psi(x) = \begin{cases} \frac{x}{e^x-1} & x\neq 0 \\ 1 & x=0 \end{cases}$, $u(x,t) = \psi(x)e^{tx}$, and for all $n \in \mathbb{N}$, let $A_{n}$ be the function from $\mathbb{R}$ to $\mathbb{R}$ such that, for all $t \in \mathbb{R}$, $$A_{n}(t) = \frac{\partial^{n} u}{\partial x^{n}}(0,t).$$ Show that, for all $n \in \mathbb{N}$, $A_{n} = B_{n}$, where $(B_n)_{n\in\mathbb{N}}$ are the Bernoulli polynomials.
Q23 Complex numbers 2 Inequalities and Estimates for Complex Expressions View
We propose to show by contradiction the property $\mathcal{P}$: $$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right).$$ We suppose that $\mathcal{P}$ is false. Show the existence of a sequence of natural integers $(n_{p})_{p \in \mathbb{N}}$ and a sequence of complex numbers $(z_{p})_{p \in \mathbb{N}}$ such that: $$\mathrm{e}^{z_{p}} \underset{p \rightarrow +\infty}{\rightarrow} 1 \quad \text{and} \quad \forall p \in \mathbb{N},\, |z_{p}| = (2n_{p}+1)\pi.$$
Q24 Complex numbers 2 Inequalities and Estimates for Complex Expressions View
We suppose that the property $\mathcal{P}$: $$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right)$$ is false, and let $(n_p)_{p\in\mathbb{N}}$, $(z_p)_{p\in\mathbb{N}}$ be sequences such that $\mathrm{e}^{z_p} \to 1$ and $|z_p| = (2n_p+1)\pi$ for all $p$. For all $p \in \mathbb{N}$, we denote $a_{p} = \operatorname{Re}(z_{p})$ and $b_{p} = \operatorname{Im}(z_{p})$. Show that $a_{p} \underset{p \rightarrow +\infty}{\rightarrow} 0$ and $|z_{p}| - |b_{p}| \underset{p \rightarrow +\infty}{\rightarrow} 0$.
Q25 Complex numbers 2 Inequalities and Estimates for Complex Expressions View
We suppose that the property $\mathcal{P}$: $$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right)$$ is false, and let $(n_p)_{p\in\mathbb{N}}$, $(z_p)_{p\in\mathbb{N}}$ be sequences such that $\mathrm{e}^{z_p} \to 1$ and $|z_p| = (2n_p+1)\pi$ for all $p$, with $a_p = \operatorname{Re}(z_p)$, $b_p = \operatorname{Im}(z_p)$. For all $p \in \mathbb{N}$, we denote $$\varepsilon_{p} = \begin{cases} +1 & \text{if } b_{p} \geqslant 0 \\ -1 & \text{if } b_{p} < 0 \end{cases}$$ By studying $\exp(z_{p} - \mathrm{i}\varepsilon_{p}|z_{p}|)$, reach a contradiction and conclude that $\mathcal{P}$ is true.
Q26 Complex numbers 2 Contour Integration and Residue Calculus View
We denote by $\mathcal{E}$ the set of functions $f : \mathbb{C} \rightarrow \mathbb{C}$ expandable as a power series with radius of convergence infinity. For all $n \in \mathbb{N}$, we define $$\gamma_{n} : \begin{cases} [0,1] \rightarrow \mathbb{C} \\ t \mapsto (2n+1)\pi\, \mathrm{e}^{2\mathrm{i}\pi t} \end{cases}$$ and for all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, $$Q_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\gamma_{n}(t)}}{(\mathrm{e}^{\gamma_{n}(t)} - 1)\gamma_{n}(t)^{n-1}} \,\mathrm{d}t.$$ Show that, for all $n \in \mathbb{N}$, $Q_{n} \in \mathcal{E}$.
Q27 Complex numbers 2 Contour Integration and Residue Calculus View
For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, let $$Q_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\gamma_{n}(t)}}{(\mathrm{e}^{\gamma_{n}(t)} - 1)\gamma_{n}(t)^{n-1}} \,\mathrm{d}t$$ where $\gamma_{n}(t) = (2n+1)\pi\,\mathrm{e}^{2\mathrm{i}\pi t}$. Show that $$\forall n \in \mathbb{N}^{*},\, \forall z \in \mathbb{C}, \quad Q_{n}(z+1) - Q_{n}(z) = n z^{n-1}.$$
Q28 Complex numbers 2 Inequalities and Estimates for Complex Expressions View
For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, let $$Q_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\gamma_{n}(t)}}{(\mathrm{e}^{\gamma_{n}(t)} - 1)\gamma_{n}(t)^{n-1}} \,\mathrm{d}t$$ where $\gamma_{n}(t) = (2n+1)\pi\,\mathrm{e}^{2\mathrm{i}\pi t}$. Using the property $\mathcal{P}$: $$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right),$$ show that there exist two constants $a, b \in \mathbb{R}_{+}^{*}$ such that, for all $n \in \mathbb{N}^{*}$ and all $z \in \mathbb{C}$, $$|Q_{n}(z)| \leqslant a\,\mathrm{e}^{bn|z|}.$$
Q29 Sequences and series, recurrence and convergence Series convergence and power series analysis View
We denote by $\mathcal{E}$ the set of functions $f : \mathbb{C} \rightarrow \mathbb{C}$ expandable as a power series with radius of convergence infinity. Using the functions $Q_n \in \mathcal{E}$ satisfying $Q_n(z+1) - Q_n(z) = nz^{n-1}$ for all $n \in \mathbb{N}^*$ and $z \in \mathbb{C}$, and the bound $|Q_n(z)| \leqslant a\,\mathrm{e}^{bn|z|}$ for constants $a,b \in \mathbb{R}_+^*$, deduce the existence of a solution in $\mathcal{E}$ to the equation $(E_h)$: $$\forall z \in \mathbb{C},\, f(z+1) - f(z) = h(z)$$ when $h \in \mathcal{E}$.