grandes-ecoles 2016 QI.A.3

grandes-ecoles · France · centrale-maths1__pc Reduction Formulae Establish an Integral Identity or Representation

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Show the existence of the two integrals $\int_{0}^{+\infty} e^{-t^{2}} \mathrm{~d}t$ and $\int_{0}^{+\infty} e^{-t^{4}} \mathrm{~d}t$ and express them using $\Gamma$.

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Show the existence of the two integrals $\int_{0}^{+\infty} e^{-t^{2}} \mathrm{~d}t$ and $\int_{0}^{+\infty} e^{-t^{4}} \mathrm{~d}t$ and express them using $\Gamma$.

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