grandes-ecoles 2016 QII.B.3

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

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$. Let $R(x)$ be the real part and $I(x)$ be the imaginary part of $F(x)$.
Determine, in a neighbourhood of 0, the Taylor expansion of $R(x)$ to order 3 and of $I(x)$ to order 4.

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$. Let $R(x)$ be the real part and $I(x)$ be the imaginary part of $F(x)$.

Determine, in a neighbourhood of 0, the Taylor expansion of $R(x)$ to order 3 and of $I(x)$ to order 4.

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