UFM Pure

grandes-ecoles 2016 QV.C Prove smoothness or power series expandability of a function View

Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. Let $h$ be the function defined on $\mathbb{R}$, which is 1-periodic and which equals $\mathcal{F}(f)$ on the interval $[-1/2, 1/2]$.
Using the inequality from IV.G, prove the existence of a sequence of complex numbers $(d_{k})_{k \in \mathbb{Z}}$ such that the sequence of functions $\left(x \mapsto \sum_{k=-n}^{n} d_{k} e^{2\pi\mathrm{i} kx}\right)_{n \in \mathbb{N}}$ converges uniformly to $\mathcal{F}(f)$ on $[-1/2, 1/2]$.

grandes-ecoles 2016 QV.D Prove smoothness or power series expandability of a function View

Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. We set
$$\forall k \in \mathbb{Z}, \quad \forall x \in \mathbb{R}, \quad \psi_{k}(x) = \psi(x+k)$$
where $\psi(x) = \frac{\sin(\pi x)}{\pi x}$ for $x \neq 0$ and $\psi(0) = 1$.
Let $(d_{k})_{k \in \mathbb{Z}}$ be the sequence of complex numbers from V.C. Prove that the sequence of functions $\left(\sum_{k=-n}^{n} d_{k} \psi_{k}\right)_{n \in \mathbb{N}}$ converges uniformly to $f$ on $\mathbb{R}$.

grandes-ecoles 2016 QV.E Prove smoothness or power series expandability of a function View

Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. We set
$$\forall k \in \mathbb{Z}, \quad \forall x \in \mathbb{R}, \quad \psi_{k}(x) = \psi(x+k)$$
where $\psi(x) = \frac{\sin(\pi x)}{\pi x}$ for $x \neq 0$ and $\psi(0) = 1$, and $f = \sum_{k=-\infty}^{+\infty} d_{k} \psi_{k}$ (uniform limit).
Establish that $\forall j \in \mathbb{Z},\ f(-j) = d_{j}$.

grandes-ecoles 2017 QII.A.1 Construct series for a composite or related function View

We recall that the hyperbolic cosine function, which we denote cosh, is defined, for every real $t$, by $$\cosh(t)=\frac{\mathrm{e}^{t}+\mathrm{e}^{-t}}{2}$$
a) Give the power series expansion of the hyperbolic cosine function and that of the function defined on $\mathbb{R}$ by $t \mapsto \mathrm{e}^{t^{2}/2}$. We will give the radius of convergence of these two power series.
b) Deduce that $\forall t \in \mathbb{R}, \cosh(t) \leqslant \mathrm{e}^{t^{2}/2}$.

grandes-ecoles 2018 QIV.4 Taylor's formula with integral remainder or asymptotic expansion View

Using the results of the previous questions (in particular the integral representation of $I_n$ from question 2, the bounds from question 3, and the Gaussian integral $\int_{-\infty}^{+\infty} e^{-x^{2}/2}\, dx = \sqrt{2\pi}$), deduce Stirling's formula: $$n! \underset{n \rightarrow \infty}{\sim} \sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}.$$

grandes-ecoles 2018 Q3 Extract derivative values from a given series View

We assume that, for all non-zero natural integer $n$, $X$ admits a moment of order $n$ and that the power series $\sum _ { n \geqslant 0 } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$ has a radius of convergence $R _ { X } > 0$. For all $t \in ] - R _ { X } , R _ { X } [$, we denote $M _ { X } ( t ) = \sum _ { n = 0 } ^ { + \infty } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$.
Justify that knowledge of the function $M _ { X }$ allows us to determine uniquely the sequence $\left( m _ { n } ( X ) \right) _ { n \in \mathbb { N } ^ { * } }$.

grandes-ecoles 2018 Q10 Formal power series manipulation (Cauchy product, algebraic identities) View

Show that, for any real $\xi$, there exists a real sequence $\left(c_{p}(\xi)\right)_{p \in \mathbb{N}}$ such that $$\forall x \in \mathbb{R}, \quad \exp\left(-x^{2}\right) \cos(2\pi \xi x) = \sum_{p=0}^{+\infty} c_{p}(\xi) \exp\left(-x^{2}\right) x^{2p}$$

grandes-ecoles 2018 Q12 Taylor's formula with integral remainder or asymptotic expansion View

Let $n \in \mathbb{N}^*$, $h = \frac{1}{n+1}$, and $x_i = ih$ for all $i \in \{0, \ldots, n+1\}$. Show that for any function $v \in \mathcal { C } ^ { 4 } ( [ 0,1 ] , \mathbb { R } )$, there exists a constant $C \geq 0$, independent of $n$, such that
$$\forall i \in \{ 1 , \ldots , n \} , \left| v ^ { \prime \prime } \left( x _ { i } \right) - \frac { 1 } { h ^ { 2 } } \left( v \left( x _ { i + 1 } \right) + v \left( x _ { i - 1 } \right) - 2 v \left( x _ { i } \right) \right) \right| \leq C h ^ { 2 }$$

grandes-ecoles 2018 Q18 Prove smoothness or power series expandability of a function View

We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Deduce that $\varphi$ is of class $C ^ { \infty }$ on $\mathbb { R }$ and for $p \in \mathbb { N } ^ { * }$, give the value of $\varphi ^ { ( p ) } ( 1 )$.

grandes-ecoles 2018 Q19 Identify a closed-form function from its Taylor series View

Using question 15, determine, for any real $\xi$, the value of $K(\xi)$.

grandes-ecoles 2018 Q19 Formal power series manipulation (Cauchy product, algebraic identities) View

We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Demonstrate, for all $x \in ] - 1,1 [$, $$\varphi ( x ) = \sum _ { q = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { q } } { q ! } x ^ { q } ( 1 - x ) ^ { - q / 2 }$$

grandes-ecoles 2018 Q20 Construct series for a composite or related function View

Deduce the existence of a real $\nu_{\sigma}$ such that, for any real $\xi$ and any real $t > 0$, $$\hat{f}(t, \xi) = \nu_{\sigma} \exp\left(-2\pi^{2}\left(\sigma^{2}+2t\right) \xi^{2}\right)$$

grandes-ecoles 2018 Q21 Extract derivative values from a given series View

Give the value of $\nu_{\sigma}$.

grandes-ecoles 2018 Q28 Prove smoothness or power series expandability of a function View

A function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands as 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 as 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 as a power series on $D(0,R)$.

grandes-ecoles 2018 Q28 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 $D(0,R)$.

grandes-ecoles 2018 Q29 Prove smoothness or power series expandability of a function View

A function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands as 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 $D(0,R)$.

grandes-ecoles 2018 Q29 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 $D(0,R)$.

grandes-ecoles 2018 Q30 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 as a power series on $D(0,R)$.

grandes-ecoles 2018 Q30 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 $D(0,R)$.

grandes-ecoles 2018 Q31 Prove smoothness or power series expandability of a function View

Show that if $g$ belongs to $\mathcal{H}(D(0,R))$ then there exists a function $H$ that expands as 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.

grandes-ecoles 2018 Q31 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.

grandes-ecoles 2018 Q32 Formal power series manipulation (Cauchy product, algebraic identities) View

Let $f$ be a function that expands as a power series on $D(0,R)$, i.e., there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i}y)^n$$ Show that for all $r \in [0, R[$, we have $f(0) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos(t), r\sin(t))\, \mathrm{d}t$.

grandes-ecoles 2018 Q32 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 $f(0) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos(t), r\sin(t)) \, \mathrm{d}t$.

grandes-ecoles 2018 Q33 Prove smoothness or power series expandability of a function View

Show an analogous result to $f(0) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos(t), r\sin(t))\, \mathrm{d}t$ for harmonic functions.

grandes-ecoles 2018 Q33 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 $r \in [0, R[$, $g(0) = \frac{1}{2\pi} \int_0^{2\pi} g(r\cos(t), r\sin(t)) \, \mathrm{d}t$.

UFM Pure

Sequences and series, recurrence and convergence 375

Roots of polynomials 146

Polar coordinates 22

Conic sections 291

Taylor series 183

Hyperbolic functions 10

Integration using inverse trig and hyperbolic functions 6

Integration with Partial Fractions 7

Vectors: Lines & Planes 293

Vectors: Cross Product & Distances 20

First order differential equations (integrating factor) 60

Complex numbers 2 113

Second order differential equations 127

Systems of differential equations 41