grandes-ecoles 2016 QI.C.2

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

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Deduce the variations of $\Gamma$ on $\mathcal{D}$. Specify in particular the limits of $\Gamma$ at 0 and at $+\infty$. Also specify the limits of $\Gamma^{\prime}$ at 0 and at $+\infty$. Sketch the graph of $\Gamma$.

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Deduce the variations of $\Gamma$ on $\mathcal{D}$. Specify in particular the limits of $\Gamma$ at 0 and at $+\infty$. Also specify the limits of $\Gamma^{\prime}$ at 0 and at $+\infty$. Sketch the graph of $\Gamma$.

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