%BECHATA Abdellah
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{\Large ESG 1989 Option économique et technologique\vspace{2cm}}
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\section*{EXERCICE\ N$°$ I}
\begin{enumerate}
\item Dire pourquoi les limites $\int\limits_{-A}^{B}x^{k}e^{-x^{2}}dx$
existent pour $k>0$ fixé, quand $A$ et $B$ tendent vers $+\infty .$\newline
On notera $\int\limits_{-\infty }^{+\infty }x^{k}e^{-x^{2}}dx$ cette limite.
\item Montrer que la dérivée $\dfrac{d^{n}}{dt^{n}}(e^{-t^{2}})$ est de la
forme $(-1)^{n}H_{n}(t)e^{-t^{2}}$ où $H_{n}$ est un polynôme de degré $n,$
montrer que $H_{0}=1$ et $H_{1}(t)=2t$ et
\begin{equation*}
H_{n+1}(t)=2tH_{n}(t)-H_{n}^{\prime }(t)\qquad (1)
\end{equation*}%
Montrer que $H_{n}$ est pair ou impair suivant $n.$
\item Montrer que $\int\limits_{-\infty }^{+\infty
}H_{n}(t)e^{-t^{2}}t^{k}dt=0$ dès que $k