grandes-ecoles 2016 QI.A.2

grandes-ecoles · France · centrale-maths1__pc Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. For all $x \in \mathcal{D}$, express $\Gamma(x+1)$ in terms of $x$ and $\Gamma(x)$.
Deduce from this, for all $x \in \mathcal{D}$ and all $n \in \mathbb{N}^{*}$, an expression for $\Gamma(x+n)$ in terms of $x$, $n$ and $\Gamma(x)$, as well as the value of $\Gamma(n)$ for all $n \geqslant 1$.

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. For all $x \in \mathcal{D}$, express $\Gamma(x+1)$ in terms of $x$ and $\Gamma(x)$.

Deduce from this, for all $x \in \mathcal{D}$ and all $n \in \mathbb{N}^{*}$, an expression for $\Gamma(x+n)$ in terms of $x$, $n$ and $\Gamma(x)$, as well as the value of $\Gamma(n)$ for all $n \geqslant 1$.

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