grandes-ecoles 2015 QII.B.4

grandes-ecoles · France · centrale-maths2__mp Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$.
Deduce that for $x > 0 , y > 0 , \beta ( x + 1 , y + 1 ) = \frac { x y } { ( x + y ) ( x + y + 1 ) } \beta ( x , y )$.

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$.

Deduce that for $x > 0 , y > 0 , \beta ( x + 1 , y + 1 ) = \frac { x y } { ( x + y ) ( x + y + 1 ) } \beta ( x , y )$.

Paper Questions

QI.A.1 QI.A.2 QI.B QI.C.1 QI.C.2 QI.D.1 QI.D.2 QI.D.3 QI.D.4 QI.E QI.F.1 QI.F.2 QI.F.3 QII.A.1 QII.A.2 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QII.C.1 QII.C.2 QII.C.3 QII.C.4 QII.C.5 QII.C.6 QII.C.7 QII.C.8 QIII.A QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIII.C.3 QIII.D.1 QIII.D.2 QIII.D.3 QIV.A QIV.B.1 QIV.B.2 QIV.B.3 QIV.B.4 QIV.C.1 QIV.C.2