grandes-ecoles 2011 QIV.B.2

grandes-ecoles · France · centrale-maths1__mp Sequences and Series Functional Equations and Identities via Series

For every natural integer $p \geqslant 1$ and every real number $x$, we set $\widetilde { A } _ { p } ( x ) = A _ { p } \left( \frac { x } { 2 \pi } - \left[ \frac { x } { 2 \pi } \right] \right)$. Using question III.B.1, determine the Fourier coefficients of $\widetilde { A } _ { p }$: $$\widehat { A } _ { p } ( n ) = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \widetilde { A } _ { p } ( t ) e ^ { - i n x } d x$$

For every natural integer $p \geqslant 1$ and every real number $x$, we set $\widetilde { A } _ { p } ( x ) = A _ { p } \left( \frac { x } { 2 \pi } - \left[ \frac { x } { 2 \pi } \right] \right)$. Using question III.B.1, determine the Fourier coefficients of $\widetilde { A } _ { p }$:
$$\widehat { A } _ { p } ( n ) = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \widetilde { A } _ { p } ( t ) e ^ { - i n x } d x$$

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.B.1 QI.B.2 QI.B.3 QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QII.B.3 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIV.A.1 QIV.A.2 QIV.A.3 QIV.B.1 QIV.B.2 QIV.B.3 QIV.B.4 QIV.C.1 QIV.C.2