grandes-ecoles 2016 QII.B.1

grandes-ecoles · France · centrale-maths1__pc Taylor series Taylor's formula with integral remainder or asymptotic expansion

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$. Show that in a neighbourhood of $x = 0$, the function $F$ can be written in the form
$$F(x) = \sum_{n=0}^{+\infty} c_{n} \frac{(\mathrm{i}x)^{n}}{n!} \tag{S}$$
where $c_{n}$ is the value of Gamma at a point to be specified. Express $c_{n}$ in terms of $n$ and $c_{0}$.
What is the radius of convergence of the power series appearing on the right-hand side of $(S)$?

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$. Show that in a neighbourhood of $x = 0$, the function $F$ can be written in the form

$$F(x) = \sum_{n=0}^{+\infty} c_{n} \frac{(\mathrm{i}x)^{n}}{n!} \tag{S}$$

where $c_{n}$ is the value of Gamma at a point to be specified. Express $c_{n}$ in terms of $n$ and $c_{0}$.

What is the radius of convergence of the power series appearing on the right-hand side of $(S)$?

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