grandes-ecoles 2016 QI.B.2

grandes-ecoles · France · centrale-maths1__pc Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Show that $\Gamma$ is of class $\mathcal{C}^{\infty}$ on $\mathcal{D}$.
Let $k \in \mathbb{N}^{*}$ and $x \in \mathcal{D}$. Express $\Gamma^{(k)}(x)$, the $k$-th derivative of $\Gamma$ at point $x$, in the form of an integral.

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Show that $\Gamma$ is of class $\mathcal{C}^{\infty}$ on $\mathcal{D}$.

Let $k \in \mathbb{N}^{*}$ and $x \in \mathcal{D}$. Express $\Gamma^{(k)}(x)$, the $k$-th derivative of $\Gamma$ at point $x$, in the form of an integral.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.B.1 QI.B.2 QI.C.1 QI.C.2 QII.A QII.B.1 QII.B.2 QII.B.3 QII.C.1 QII.C.2 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIII.C.3 QIII.C.4 QIII.C.5 QIII.C.6 QIII.D.1 QIII.D.2 QIII.D.3 QIII.E.1 QIII.E.2 QIII.F