grandes-ecoles 2011 QIII.B.1

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

Let $f$ be a complex function of class $C ^ { \infty }$ on $[ 0,1 ]$.
a) Show that for every integer $q \geqslant 1$ $$f ( 1 ) - f ( 0 ) = \sum _ { j = 1 } ^ { q } ( - 1 ) ^ { j + 1 } \left[ A _ { j } ( t ) f ^ { ( j ) } ( t ) \right] _ { 0 } ^ { 1 } + ( - 1 ) ^ { q } \int _ { 0 } ^ { 1 } A _ { q } ( t ) f ^ { ( q + 1 ) } ( t ) d t$$
b) Taking into account the relations found in the previous part, show that for every odd natural integer $q = 2 p + 1$ $$f ( 1 ) - f ( 0 ) = \frac { 1 } { 2 } \left( f ^ { \prime } ( 0 ) + f ^ { \prime } ( 1 ) \right) - \sum _ { j = 1 } ^ { p } a _ { 2 j } \left( f ^ { ( 2 j ) } ( 1 ) - f ^ { ( 2 j ) } ( 0 ) \right) - \int _ { 0 } ^ { 1 } A _ { 2 p + 1 } ( t ) f ^ { ( 2 p + 2 ) } ( t ) d t$$

Let $f$ be a complex function of class $C ^ { \infty }$ on $[ 0,1 ]$.

a) Show that for every integer $q \geqslant 1$
$$f ( 1 ) - f ( 0 ) = \sum _ { j = 1 } ^ { q } ( - 1 ) ^ { j + 1 } \left[ A _ { j } ( t ) f ^ { ( j ) } ( t ) \right] _ { 0 } ^ { 1 } + ( - 1 ) ^ { q } \int _ { 0 } ^ { 1 } A _ { q } ( t ) f ^ { ( q + 1 ) } ( t ) d t$$

b) Taking into account the relations found in the previous part, show that for every odd natural integer $q = 2 p + 1$
$$f ( 1 ) - f ( 0 ) = \frac { 1 } { 2 } \left( f ^ { \prime } ( 0 ) + f ^ { \prime } ( 1 ) \right) - \sum _ { j = 1 } ^ { p } a _ { 2 j } \left( f ^ { ( 2 j ) } ( 1 ) - f ^ { ( 2 j ) } ( 0 ) \right) - \int _ { 0 } ^ { 1 } A _ { 2 p + 1 } ( t ) f ^ { ( 2 p + 2 ) } ( t ) d t$$

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