grandes-ecoles

Q1 Proof Proof of Set Membership, Containment, or Structural Property View

Show that $\mathcal{H}(U)$ is a vector subspace of $\mathcal{C}^2(U, \mathbb{R})$.

Q2 Proof Proof of Stability or Invariance View

Let $f \in \mathcal{H}(U)$. Show that if $f$ is $\mathcal{C}^\infty$ on $U$, then every partial derivative of any order of $f$ belongs to $\mathcal{H}(U)$.

Q3 Proof Characterization or Determination of a Set or Class View

We assume in this question that $U$ is path-connected. Determine the set of functions $f$ in $\mathcal{H}(U)$ such that $f^2$ also belongs to $\mathcal{H}(U)$.

Q4 Differential equations Higher-Order and Special DEs (Proof/Theory) View

Give a non-constant function belonging to $\mathcal{H}(U)$. Is the product of two harmonic functions a harmonic function?

Q5 Differential equations Higher-Order and Special DEs (Proof/Theory) View

We seek to determine the non-zero harmonic functions on $\mathbb{R}^2$ with separable variables, that is, functions $f$ that can be written in the form $f(x,y) = u(x)v(y)$. We are given two functions $u$ and $v$, of class $\mathcal{C}^2$ on $\mathbb{R}$, not identically zero, and we set $$\forall (x,y) \in \mathbb{R}^2, \quad f(x,y) = u(x)v(y)$$ We assume that $f$ is harmonic on $\mathbb{R}^2$.
Show that there exists a real constant $\lambda$ such that $u$ and $v$ are solutions respectively of the equations $$z'' + \lambda z = 0 \quad \text{and} \quad z'' - \lambda z = 0$$

Q6 Differential equations First-Order Linear DE: General Solution View

We seek to determine the non-zero harmonic functions on $\mathbb{R}^2$ with separable variables, that is, functions $f$ that can be written in the form $f(x,y) = u(x)v(y)$. We are given two functions $u$ and $v$, of class $\mathcal{C}^2$ on $\mathbb{R}$, not identically zero, and we set $$\forall (x,y) \in \mathbb{R}^2, \quad f(x,y) = u(x)v(y)$$ We assume that $f$ is harmonic on $\mathbb{R}^2$.
Give, depending on the sign of $\lambda$, the form of harmonic functions with separable variables.

Q7 Connected Rates of Change Partial Derivative Coordinate Transformation View

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Justify that $g$ is of class $\mathcal{C}^2$ on $\mathbb{R}^{*+} \times \mathbb{R}$.

Q8 Connected Rates of Change Partial Derivative Coordinate Transformation View

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ For all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, express $\frac{\partial g}{\partial r}(r,\theta)$ and $\frac{\partial g}{\partial \theta}(r,\theta)$ in terms of $$\frac{\partial f}{\partial x}(r\cos(\theta), r\sin(\theta)) \quad \text{and} \quad \frac{\partial f}{\partial y}(r\cos(\theta), r\sin(\theta))$$

Q9 Connected Rates of Change Partial Derivative Coordinate Transformation View

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Also express $\frac{\partial^2 g}{\partial r^2}(r,\theta)$ and $\frac{\partial^2 g}{\partial \theta^2}(r,\theta)$ in terms of the first and second partial derivatives of $f$ at $(r\cos(\theta), r\sin(\theta))$.

Q10 Differential equations Higher-Order and Special DEs (Proof/Theory) View

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Show that $f$ belongs to $\mathcal{H}(\mathbb{R}^2 \setminus \{(0,0)\})$ if and only if, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$r^2 \frac{\partial^2 g}{\partial r^2}(r,\theta) + \frac{\partial^2 g}{\partial \theta^2}(r,\theta) + r\frac{\partial g}{\partial r}(r,\theta) = 0$$

Q11 Differential equations First-Order Linear DE: General Solution View

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Determine the radial harmonic functions of $\mathbb{R}^2 \setminus \{(0,0)\}$, that is, the functions $f$ belonging to $\mathcal{H}(\mathbb{R}^2 \setminus \{(0,0)\})$ such that $(r,\theta) \mapsto f(r\cos(\theta), r\sin(\theta))$ is independent of $\theta$.

Q12 Differential equations First-Order Linear DE: General Solution View

Let $a, b, r_1$ and $r_2$ be four real numbers such that $0 < r_1 < r_2$. Determine a function $f$ of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$ such that $$\begin{cases} \Delta f = 0 \\ f(x,y) = a & \text{if } \|(x,y)\| = r_1 \\ f(x,y) = b & \text{if } \|(x,y)\| = r_2 \end{cases}$$

Q13 Differential equations Higher-Order and Special DEs (Proof/Theory) View

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ The function $f$ is then a function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$, called a function with separable polar variables.
Show that, if $f$ is not identically zero, then $v$ is $2\pi$-periodic.

Q14 Differential equations Higher-Order and Special DEs (Proof/Theory) View

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ The function $f$ is then a function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$, called a function with separable polar variables.
Show that, if $f$ is harmonic and not identically zero on $\mathbb{R}^2 \setminus \{(0,0)\}$, then there exists a real number $\lambda$ such that $u$ is a solution of the differential equation (II.1) $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$ and $v$ is a solution of the differential equation (II.2) $$z''(\theta) + \lambda z(\theta) = 0$$

Q15 Differential equations First-Order Linear DE: General Solution View

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We assume here that $\lambda = 0$. What are the $2\pi$-periodic solutions of (II.2): $$z''(\theta) + \lambda z(\theta) = 0$$

Q16 Differential equations First-Order Linear DE: General Solution View

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We assume here that $\lambda = 0$. Solve (II.1) on $\mathbb{R}^{+*}$: $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$

Q17 Differential equations Higher-Order and Special DEs (Proof/Theory) View

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We assume here that $\lambda = 0$. Deduce from this, in the case $\lambda = 0$, the harmonic functions with separable polar variables.

Q18 Second order differential equations Second-order ODE with initial or boundary value conditions View

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We now assume $\lambda \neq 0$. Give a necessary and sufficient condition for (II.2) $$z''(\theta) + \lambda z(\theta) = 0$$ to admit non-zero $2\pi$-periodic solutions. Give these solutions.

Q19 Second order differential equations Euler-type (Cauchy-Euler) second-order ODE View

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We now assume $\lambda \neq 0$. Solve (II.1) on $\mathbb{R}^{+*}$: $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$ One may consider, justifying its existence, a function $Z$ of class $\mathcal{C}^2$ on $\mathbb{R}$ such that, for all $r > 0$, $z(r) = Z(\ln(r))$.

Q20 Second order differential equations Qualitative and asymptotic analysis of solutions View

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We now assume $\lambda \neq 0$. Which are the solutions of (II.1) that extend continuously to 0?

Q21 Stationary points and optimisation Existence or properties of extrema via abstract/theoretical argument View

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$) and $f : U \rightarrow \mathbb{R}$ of class $\mathcal{C}^2$. Let $f$ be a function continuous on $\bar{U}$. Show that $f$ attains a maximum at some point $x_0 \in \bar{U}$.

Q22 Applied differentiation Partial derivatives and multivariable differentiation View

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$) and $f : U \rightarrow \mathbb{R}$ of class $\mathcal{C}^2$. Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ on $U$, and such that for all $x \in U$, $\Delta f(x) > 0$. Show that $x_0 \in \partial U$ and deduce that $\forall x \in U, f(x) < \sup_{y \in \partial U} f(y)$.
One may assume by contradiction that $x_0 \in U$, justify that there exists $i \in \llbracket 1, n \rrbracket$ such that $\frac{\partial^2 f}{\partial x_i^2}(x_0) > 0$, and consider the function $\varphi$ defined, for $t$ real, by $\varphi(t) = f(x_0 + t e_i)$, where $e_i$ denotes the $i$-th vector of the canonical basis of $\mathbb{R}^n$.

Q23 Applied differentiation Partial derivatives and multivariable differentiation View

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$). Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ and harmonic on $U$. For all $\varepsilon > 0$ we set $g_\varepsilon(x) = f(x) + \varepsilon \|x\|^2$. Show that $g_\varepsilon$ is a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ on $U$, and such that $\forall x \in U, \Delta g_\varepsilon(x) > 0$.

Q24 Proof Deduction or Consequence from Prior Results View

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$). Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ and harmonic on $U$. For all $\varepsilon > 0$ we set $g_\varepsilon(x) = f(x) + \varepsilon \|x\|^2$. Deduce that $\forall x \in U, f(x) \leqslant \sup_{y \in \partial U} f(y)$.

Q25 Proof Direct Proof of a Stated Identity or Equality View

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$). Let $f_1$ and $f_2$ be two functions continuous on $\bar{U}$, of class $\mathcal{C}^2$ and harmonic on $U$. Show that if the functions $f_1$ and $f_2$ are equal on $\partial U$, then $f_1$ and $f_2$ are equal on $U$.

Q26 Sequences and series, recurrence and convergence Series convergence and power series analysis View

We say that a function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands in a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ Throughout this part, $f$ denotes a function that expands in a power series on $D(0,R)$.
Show that $f$ is of class $\mathcal{C}^1$ on $D(0,R)$ and that its partial derivatives expand in power series on $D(0,R)$. What can we deduce about the function $f$?

Q27 Sequences and series, recurrence and convergence Series convergence and power series analysis View

We say that a function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands in a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ We denote by $u$ and $v$ the real and imaginary parts of $f$, so that, for any $(x,y) \in D(0,R)$, $$u(x,y) \in \mathbb{R}, \quad v(x,y) \in \mathbb{R}, \quad f(x,y) = u(x,y) + \mathrm{i} v(x,y).$$ Show that $u$ and $v$ are harmonic functions on $D(0,R)$.

Q28 Taylor series Prove smoothness or power series expandability of a function View

We say that a function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands in a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ We admit the following result: a function $h$ from $D(0,R)$ to $\mathbb{C}$ expands in a power series on $D(0,R)$ if and only if $h$ is of class $\mathcal{C}^1$ on $D(0,R)$ and for all $(x,y) \in D(0,R)$, $\frac{\partial h}{\partial y}(x,y) = \mathrm{i} \frac{\partial h}{\partial x}(x,y)$.
Show that if $f$ does not vanish on $D(0,R)$ then $1/f$ expands in a power series on $D(0,R)$.

Q29 Taylor series Prove smoothness or power series expandability of a function View

We say that a function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands in a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ We denote by $u$ and $v$ the real and imaginary parts of $f$. Show that the function $uv$ is harmonic on $D(0,R)$.

Q30 Taylor series Prove smoothness or power series expandability of a function View

Let $g$ be a function from $D(0,R) \subset \mathbb{R}^2$ to $\mathbb{R}$. We assume that $g$ is harmonic. Show that the function $h$ defined on $D(0,R)$ by $$h : (x,y) \longmapsto \frac{\partial g}{\partial x}(x,y) - \mathrm{i} \frac{\partial g}{\partial y}(x,y)$$ expands in a power series on $D(0,R)$.

Q31 Taylor series Prove smoothness or power series expandability of a function View

Let $g$ be a function from $D(0,R) \subset \mathbb{R}^2$ to $\mathbb{R}$. We assume that $g$ is harmonic. Show that if $g$ belongs to $\mathcal{H}(D(0,R))$ then there exists a function $H$ that expands in a power series on $D(0,R)$ such that $g$ is the real part of $H$.
One may consider a power series that is a primitive of the power series associated with the function $h$ from the previous question.

Q32 Taylor series Formal power series manipulation (Cauchy product, algebraic identities) View

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 for all $r \in [0, R[$, we have $f(0) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos(t), r\sin(t)) \, \mathrm{d}t$.

Q33 Taylor series Prove smoothness or power series expandability of a function View

Show an analogous result to Q32 for harmonic functions: for a harmonic function $g$ on $D(0,R)$, show that for all $r \in [0, R[$, $g(0) = \frac{1}{2\pi} \int_0^{2\pi} g(r\cos(t), r\sin(t)) \, \mathrm{d}t$.

Q34 Taylor series Prove smoothness or power series expandability of a function View

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 $\forall r \in [0, R[$, $|f(0)| \leqslant \sup_{t \in \mathbb{R}} |f(r\cos(t), r\sin(t))|$.

Q35 Taylor series Prove smoothness or power series expandability of a function View

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

Q36 Complex numbers 2 Properties of Analytic/Entire Functions View

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 if $|f|$ attains a maximum at 0, then $f$ is constant on $D(0,R)$.

Q37 Complex numbers 2 Properties of Analytic/Entire Functions View

Show the d'Alembert-Gauss theorem: every non-constant complex polynomial has at least one root.
One may proceed by contradiction, assume that there exists a polynomial that does not vanish, and consider its inverse.

Q38 Complex numbers 2 Properties of Analytic/Entire Functions View

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. We seek to solve the Dirichlet problem on the unit disk; we need to determine the function or functions $f$ defined and continuous on $\overline{D(0,1)}$, of class $\mathcal{C}^2$ on $D(0,1)$, and such that $$\begin{cases} \Delta f = 0 \text{ on } D(0,1) \\ \forall t \in \mathbb{R}, f(\cos(t), \sin(t)) = h(t) \end{cases}$$ For this, we set, for any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that the function $z \mapsto \frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}$ is expandable in a power series for $|z| < 1$ and calculate its power series expansion. Deduce that the function $(x,y) \mapsto g(x + \mathrm{i}y)$ is a harmonic function on $D(0,1)$.

Q39 Complex numbers 2 Contour Integration and Residue Calculus View

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that, for any complex number $z$ such that $|z| < 1$, $\frac{1}{2\pi} \int_0^{2\pi} \mathcal{P}(t,z) \, \mathrm{d}t = 1$.

Q40 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Let $\varphi \in \mathbb{R}$. Show that, for any complex number $z$ such that $|z| < 1$, $g(z) = \frac{1}{2\pi} \int_{\varphi}^{\varphi + 2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t$.

Q41 Complex numbers 2 Contour Integration and Residue Calculus View

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that, for all $r \in [0,1[$ and all real $t$ and $\theta$, $$\mathcal{P}\left(t, r\mathrm{e}^{\mathrm{i}\theta}\right) = \frac{1 - r^2}{1 - 2r\cos(t-\theta) + r^2}$$

Q42 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that, for all $\delta \in ]0, \pi[$ and all real $\varphi$, $$\int_{\varphi+\delta}^{\varphi+2\pi-\delta} \mathcal{P}(t,z) \, \mathrm{d}t \xrightarrow[z \rightarrow \mathrm{e}^{\mathrm{i}\varphi}]{} 0$$

Q43 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Using Heine's theorem, show that, for all $\varepsilon > 0$, there exists $\delta > 0$ such that, for all real number $\varphi$ and all complex number $z$ satisfying $|z| < 1$, $$|g(z) - h(\varphi)| \leqslant \frac{\sup_{t \in \mathbb{R}} |h(t)|}{\pi} \int_{\varphi+\delta}^{\varphi+2\pi-\delta} \mathcal{P}(t,z) \, \mathrm{d}t + \varepsilon$$

Q44 Indefinite & Definite Integrals Properties of Integral-Defined Functions (Continuity, Differentiability) View

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. We seek to solve the Dirichlet problem on the unit disk; we need to determine the function or functions $f$ defined and continuous on $\overline{D(0,1)}$, of class $\mathcal{C}^2$ on $D(0,1)$, and such that $$\begin{cases} \Delta f = 0 \text{ on } D(0,1) \\ \forall t \in \mathbb{R}, f(\cos(t), \sin(t)) = h(t) \end{cases}$$ For this, we set, for any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show the existence and uniqueness of the solution to the Dirichlet problem studied in this part.

grandes-ecoles

Papers (191)

2018 centrale-maths2__mp