grandes-ecoles

Papers (176)
2025
centrale-maths1__official 40 centrale-maths2__official 36 mines-ponts-maths1__mp 17 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 23 mines-ponts-maths2__psi 25 polytechnique-maths-a__mp 35 polytechnique-maths__fui 9 polytechnique-maths__pc 27 x-ens-maths-a__fui 10 x-ens-maths-a__mp 18 x-ens-maths-b__mp 6 x-ens-maths-c__mp 6 x-ens-maths-d__mp 31 x-ens-maths__pc 27 x-ens-maths__psi 30
2024
centrale-maths1__official 21 centrale-maths2__official 28 geipi-polytech__maths 9 mines-ponts-maths1__mp 23 mines-ponts-maths1__psi 9 mines-ponts-maths2__mp 14 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 polytechnique-maths-a__mp 42 polytechnique-maths-b__mp 27 x-ens-maths-a__mp 43 x-ens-maths-b__mp 29 x-ens-maths-c__mp 22 x-ens-maths-d__mp 41 x-ens-maths__pc 20 x-ens-maths__psi 23
2023
centrale-maths1__official 37 centrale-maths2__official 32 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 14 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 21 mines-ponts-maths2__pc 13 mines-ponts-maths2__psi 22 polytechnique-maths__fui 3 x-ens-maths-a__mp 24 x-ens-maths-b__mp 10 x-ens-maths-c__mp 10 x-ens-maths-d__mp 10 x-ens-maths__pc 22
2022
centrale-maths1__mp 22 centrale-maths1__pc 33 centrale-maths1__psi 42 centrale-maths2__mp 26 centrale-maths2__pc 37 centrale-maths2__psi 40 mines-ponts-maths1__mp 26 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 9 mines-ponts-maths2__psi 18 x-ens-maths-a__mp 8 x-ens-maths-b__mp 19 x-ens-maths-c__mp 17 x-ens-maths-d__mp 47 x-ens-maths1__mp 13 x-ens-maths2__mp 26 x-ens-maths__pc 7 x-ens-maths__pc_cpge 14 x-ens-maths__psi 22 x-ens-maths__psi_cpge 26
2021
centrale-maths1__mp 34 centrale-maths1__pc 36 centrale-maths1__psi 28 centrale-maths2__mp 21 centrale-maths2__pc 38 centrale-maths2__psi 28 x-ens-maths2__mp 35 x-ens-maths__pc 29
2020
centrale-maths1__mp 42 centrale-maths1__pc 36 centrale-maths1__psi 38 centrale-maths2__mp 2 centrale-maths2__pc 35 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 22 mines-ponts-maths2__mp_cpge 19 x-ens-maths-a__mp_cpge 10 x-ens-maths-b__mp_cpge 19 x-ens-maths-c__mp 10 x-ens-maths-d__mp 13 x-ens-maths1__mp 13 x-ens-maths2__mp 20 x-ens-maths__pc 6
2019
centrale-maths1__mp 37 centrale-maths1__pc 40 centrale-maths1__psi 38 centrale-maths2__mp 37 centrale-maths2__pc 39 centrale-maths2__psi 46 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 9
2018
centrale-maths1__mp 21 centrale-maths1__pc 31 centrale-maths1__psi 39 centrale-maths2__mp 23 centrale-maths2__pc 35 centrale-maths2__psi 30 x-ens-maths1__mp 18 x-ens-maths2__mp 13 x-ens-maths__pc 17 x-ens-maths__psi 20
2017
centrale-maths1__mp 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 24 x-ens-maths2__mp 7 x-ens-maths__pc 17 x-ens-maths__psi 19
2016
centrale-maths1__mp 41 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 42 centrale-maths2__psi 17 x-ens-maths1__mp 10 x-ens-maths2__mp 32 x-ens-maths__pc 1 x-ens-maths__psi 20
2015
centrale-maths1__mp 18 centrale-maths1__pc 11 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 1 centrale-maths2__psi 14 x-ens-maths1__mp 16 x-ens-maths2__mp 19 x-ens-maths__pc 30 x-ens-maths__psi 20
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 36 centrale-maths2__mp 24 centrale-maths2__pc 23 centrale-maths2__psi 29 x-ens-maths2__mp 13
2013
centrale-maths1__mp 3 centrale-maths1__pc 45 centrale-maths1__psi 20 centrale-maths2__mp 32 centrale-maths2__pc 50 centrale-maths2__psi 32 x-ens-maths1__mp 14 x-ens-maths2__mp 10 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__pc 23 centrale-maths1__psi 20 centrale-maths2__mp 27 centrale-maths2__psi 20
2011
centrale-maths1__mp 27 centrale-maths1__pc 15 centrale-maths1__psi 21 centrale-maths2__mp 29 centrale-maths2__pc 8 centrale-maths2__psi 28
2010
centrale-maths1__mp 7 centrale-maths1__pc 23 centrale-maths1__psi 9 centrale-maths2__mp 10 centrale-maths2__pc 36 centrale-maths2__psi 27
2018 centrale-maths2__psi

30 maths questions

Q5 Moment generating functions Existence and domain of the MGF 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 $M _ { X } ( t ) = \sum _ { n = 0 } ^ { + \infty } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$.
Show conversely that, if there exists a real $R > 0$ such that, for all $t \in ] - R , R [$, the random variable $\mathrm { e } ^ { t X }$ admits an expectation, then the domain of definition of the moment generating function of $X$ contains $] - R , R [$ and for all $t \in ] - R , R \left[ , M _ { X } ( t ) = \mathbb { E } \left( \mathrm { e } ^ { t X } \right) \right.$.
Q6 Moment generating functions MGF of sums of independent random variables (product property) View
We assume that $X$ and $Y$ are two independent discrete real-valued random variables with strictly positive values admitting moments of all orders. We denote $R _ { X }$ (respectively $R _ { Y }$) the radius of convergence (assumed strictly positive) associated with the function $M _ { X }$ (respectively $M _ { Y }$).
Show that the random variable $X + Y$ admits moments of all orders and that $$\forall | t | < \min \left( R _ { X } , R _ { Y } \right) , \quad M _ { X + Y } ( t ) = M _ { X } ( t ) \times M _ { Y } ( t )$$
Q7 Poisson distribution View
$\lambda$ is a fixed real number. We assume that $Z$ is a random variable on $(\Omega , \mathcal { A } , \mathbb { P })$ following the Poisson distribution with parameter $\lambda$.
Show that $Z$ admits moments of all orders.
Q8 Poisson distribution Compute MGF or characteristic function for a named distribution View
$\lambda$ is a fixed real number. We assume that $Z$ is a random variable on $(\Omega , \mathcal { A } , \mathbb { P })$ following the Poisson distribution with parameter $\lambda$.
Calculate the moment generating function of $Z$. Deduce the values of $m _ { 1 } ( Z )$ and $m _ { 2 } ( Z )$.
Q10 Approximating the Binomial to the Poisson distribution Explicit computation of a PGF or characteristic function View
Let $n$ be a non-zero natural integer. For $i \in \llbracket 1 , n \rrbracket$, $X _ { i }$ is a random variable on $(\Omega , \mathcal { A } , \mathbb { P })$ following a Bernoulli distribution with parameter $\lambda / n$. We assume that $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ are mutually independent and we set $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$.
For $t \in \mathbb { R }$, calculate $\lim _ { n \rightarrow + \infty } M _ { S _ { n } } ( t )$.
Q11 Approximating the Binomial to the Poisson distribution View
Let $n$ be a non-zero natural integer. For $i \in \llbracket 1 , n \rrbracket$, $X _ { i }$ is a random variable on $(\Omega , \mathcal { A } , \mathbb { P })$ following a Bernoulli distribution with parameter $\lambda / n$. We assume that $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ are mutually independent and we set $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$.
Compare with the results of question 8.
Q12 Moment generating functions Compute MGF or characteristic function for a named distribution View
For $n \in \mathbb { N } ^ { * }$, $U _ { n }$ is a random variable on $(\Omega , \mathcal { A } , \mathbb { P })$ following the uniform distribution on $\llbracket 1 , n \rrbracket$. We set $Y _ { n } = \frac { 1 } { n } U _ { n }$.
Calculate the moment generating function of the random variable $Y _ { n }$.
Q13 Moment generating functions Pointwise limit of MGFs or characteristic functions (convergence in distribution) View
For $n \in \mathbb { N } ^ { * }$, $U _ { n }$ is a random variable on $(\Omega , \mathcal { A } , \mathbb { P })$ following the uniform distribution on $\llbracket 1 , n \rrbracket$. We set $Y _ { n } = \frac { 1 } { n } U _ { n }$.
For $t \in \mathbb { R }$, calculate $\lim _ { n \rightarrow + \infty } M _ { Y _ { n } } ( t )$.
Q16 Proof Higher-order or nth derivative computation View
We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Show that, for all non-zero natural integer $p$, there exist two polynomials $P _ { p }$ and $Q _ { p }$ with real coefficients such that, for all $x \in ] - \infty , 1 [$, $$\varphi ^ { ( p ) } ( x ) = \frac { P _ { p } ( \sqrt { 1 - x } ) } { Q _ { p } ( \sqrt { 1 - x } ) } \exp \left( \frac { - x } { \sqrt { 1 - x } } \right)$$
Q17 Proof Limit involving transcendental functions 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 $\lim _ { \substack { x \rightarrow 1 \\ x < 1 } } \varphi ^ { ( p ) } ( x )$ for $p \in \mathbb { N } ^ { * }$.
Q18 Proof 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 $\varphi ^ { ( p ) } ( 1 )$.
Q19 Sequences and Series 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 $x \in ] - 1,1 [$, $$\varphi ( x ) = \sum _ { q = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { q } } { q ! } x ^ { q } ( 1 - x ) ^ { - q / 2 }$$
Q20 Sequences and Series Power Series Expansion and Radius of Convergence View
We consider the Hilbert polynomials $$\left\{ \begin{array} { l } H _ { 0 } ( X ) = 1 \\ H _ { n } ( X ) = \frac { 1 } { n ! } \prod _ { k = 0 } ^ { n - 1 } ( X - k ) = \frac { X ( X - 1 ) \cdots ( X - n + 1 ) } { n ! } \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Demonstrate that, for all $x \in ] - 1,1 [$ and all $q \in \mathbb { N } ^ { * }$, we have $$( 1 - x ) ^ { - q / 2 } = \sum _ { p = 0 } ^ { + \infty } H _ { p } \left( \frac { q } { 2 } + p - 1 \right) x ^ { p }$$
Q21 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
We consider the Hilbert polynomials $$\left\{ \begin{array} { l } H _ { 0 } ( X ) = 1 \\ H _ { n } ( X ) = \frac { 1 } { n ! } \prod _ { k = 0 } ^ { n - 1 } ( X - k ) = \frac { X ( X - 1 ) \cdots ( X - n + 1 ) } { n ! } \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Deduce $$\forall x \in ] - 1,1 \left[ , \quad \varphi ( x ) = 1 + \sum _ { i = 0 } ^ { + \infty } \left( \sum _ { j = 0 } ^ { + \infty } a _ { i , j } ( x ) \right) \right.$$ where we have set $$\forall ( i , j ) \in \mathbb { N } ^ { 2 } , \quad a _ { i , j } ( x ) = \frac { ( - 1 ) ^ { i + 1 } } { ( i + 1 ) ! } H _ { j } \left( \frac { i - 1 } { 2 } + j \right) x ^ { i + j + 1 }$$
Q22 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
We consider the Hilbert polynomials $$\left\{ \begin{array} { l } H _ { 0 } ( X ) = 1 \\ H _ { n } ( X ) = \frac { 1 } { n ! } \prod _ { k = 0 } ^ { n - 1 } ( X - k ) = \frac { X ( X - 1 ) \cdots ( X - n + 1 ) } { n ! } \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Demonstrate that, for all $x \in ] - 1,1 \left[ , \sum _ { i = 0 } ^ { + \infty } \left( \sum _ { j = 0 } ^ { + \infty } \left| a _ { i , j } ( x ) \right| \right) = \exp \left( \frac { | x | } { \sqrt { 1 - | x | } } \right) - 1 \right.$.
Q23 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
We consider the Hilbert polynomials $$\left\{ \begin{array} { l } H _ { 0 } ( X ) = 1 \\ H _ { n } ( X ) = \frac { 1 } { n ! } \prod _ { k = 0 } ^ { n - 1 } ( X - k ) = \frac { X ( X - 1 ) \cdots ( X - n + 1 ) } { n ! } \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Use the results admitted in the preamble to establish the equality $$\forall x \in ] - 1,1 \left[ , \quad \varphi ( x ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n } \right.$$ where $$\left\{ \begin{array} { l } a _ { 0 } = 1 \\ a _ { n } = \sum _ { k = 0 } ^ { n - 1 } \frac { ( - 1 ) ^ { n - k } } { ( n - k ) ! } H _ { k } \left( \frac { n + k } { 2 } - 1 \right) \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Q24 Taylor series Series convergence and power series analysis View
We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. The sequence $(a_n)$ is defined by $$\left\{ \begin{array} { l } a _ { 0 } = 1 \\ a _ { n } = \sum _ { k = 0 } ^ { n - 1 } \frac { ( - 1 ) ^ { n - k } } { ( n - k ) ! } H _ { k } \left( \frac { n + k } { 2 } - 1 \right) \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Show that, for all $z \in \mathcal { D }$, the power series $\sum _ { n \geqslant 0 } a _ { n } z ^ { n }$ converges.
Q25 Taylor series Series convergence and power series analysis View
We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. For $z \in \mathcal { D }$, we denote $\Phi _ { 0 } ( z ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } z ^ { n }$ and, subject to convergence, $$\Phi _ { p } ( z ) = \sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } z ^ { n }$$
Justify that, for all $p \in \mathbb { N } ^ { * }$ and all $\left. x \in \right] - 1,1 \left[ , \Phi _ { p } ( x ) = \varphi ^ { ( p ) } ( x ) \right.$ and that for all $p \in \mathbb { N } ^ { * }$ and all $z \in \mathcal { D }$, the power series $\sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } z ^ { n }$ converges.
Q26 Taylor series Series convergence and power series analysis View
We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Justify that, for all $p \in \mathbb { N }$, the function $\varphi ^ { ( p ) }$ is bounded on $] - 1,1 [$.
Q27 Taylor series Series convergence and power series analysis View
We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. For $z \in \mathcal { D }$, we denote $\Phi _ { p } ( z ) = \sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } z ^ { n }$. We admit that the function $\Phi _ { p }$ is bounded on $\mathcal { D }$.
Let $r$ be a real number in the interval $] 0,1 [$. Demonstrate for all integers $n \geqslant 1$ and $p \geqslant 1$, that $$( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } r ^ { n } = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \Phi _ { p } \left( r \mathrm { e } ^ { \mathrm { i } \theta } \right) \mathrm { e } ^ { - n \mathrm { i } \theta } \mathrm {~d} \theta$$
Q28 Taylor series Proof of Inequalities Involving Series or Sequence Terms View
We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. For $z \in \mathcal { D }$, we denote $\Phi _ { p } ( z ) = \sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } z ^ { n }$. We admit that the function $\Phi _ { p }$ is bounded on $\mathcal { D }$.
Demonstrate that, for all $p \in \mathbb { N }$, there exist a real $K _ { p }$ and a natural integer $N _ { p }$ such that $$\forall n \geqslant N _ { p } , \quad \left| a _ { n } \right| \leqslant \frac { K _ { p } } { n ^ { p } }$$
Q29 Taylor series Uniform or Pointwise Convergence of Function Series/Sequences View
We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. The sequence $(a_n)$ is defined by $$\left\{ \begin{array} { l } a _ { 0 } = 1 \\ a _ { n } = \sum _ { k = 0 } ^ { n - 1 } \frac { ( - 1 ) ^ { n - k } } { ( n - k ) ! } H _ { k } \left( \frac { n + k } { 2 } - 1 \right) \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Demonstrate that the power series $\sum _ { n \geqslant 0 } a _ { n } x ^ { n }$ converges normally on $[ 0,1 ]$ and give the value of $\sum _ { n = 0 } ^ { + \infty } a _ { n }$.
Q30 Taylor series Uniform or Pointwise Convergence of Function Series/Sequences View
We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. For $z \in \mathcal { D }$, we denote $\Phi _ { p } ( z ) = \sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } z ^ { n }$.
Let $p \in \mathbb { N } ^ { * }$. Show that the power series $\sum _ { n \geqslant 0 } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } x ^ { n }$ converges normally on $[ 0,1 ]$ and give the value of $\sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p }$.
Q31 Taylor series Evaluation of a Finite or Infinite Sum View
The sequence $(a_n)$ is defined by $$\left\{ \begin{array} { l } a _ { 0 } = 1 \\ a _ { n } = \sum _ { k = 0 } ^ { n - 1 } \frac { ( - 1 ) ^ { n - k } } { ( n - k ) ! } H _ { k } \left( \frac { n + k } { 2 } - 1 \right) \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Demonstrate that all moments of order $p$ of the sequence $(a _ { n })$ are zero.
Q32 Taylor series Limit Evaluation View
We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by $$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$
Show that $\lim _ { \substack { x \rightarrow 0 \\ x > 0 } } | \theta ( x ) | = 0$.
Q33 Taylor series Higher-order or nth derivative computation View
We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by $$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$
Justify that $\theta$ is of class $C ^ { \infty }$ on $\mathbb { R } ^ { + * }$ and demonstrate that, for all $n \in \mathbb { N } ^ { * }$, there exists $P _ { n } \in \mathbb { C } [ X ]$ such that $$\forall x \in ] 0 , + \infty \left[ \quad \theta ^ { ( n ) } ( x ) = \frac { P _ { n } ( \ln x ) } { x ^ { n } } \theta ( x ) \right.$$
Q34 Differentiating Transcendental Functions Limit involving transcendental functions View
We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by $$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$
Deduce that $\lim _ { \substack { x \rightarrow 0 \\ x > 0 } } \left| \theta ^ { ( n ) } ( x ) \right| = 0$. One may perform the change of variable $y = - \ln x$.
Q35 Differentiating Transcendental Functions Regularity and smoothness of transcendental functions View
We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by $$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$
Demonstrate that $\theta$ is of class $C ^ { \infty }$ on $\mathbb { R }$.
Q37 Integration by Substitution Substitution to Prove an Integral Identity or Equality View
Using the change of variable $t = \frac { \ln x } { 2 \pi }$, demonstrate that $$\forall p \in \mathbb { N } \quad I _ { p } = \frac { e ^ { - p ^ { 2 } \pi ^ { 2 } } } { 2 \pi } \int _ { 0 } ^ { + \infty } x ^ { p - 1 } \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } \right) \sin \left( \frac { \ln x } { 2 \pi } \right) \mathrm { d } x$$
Q38 Differentiating Transcendental Functions Prove smoothness or power series expandability of a function View
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.