grandes-ecoles 2015 QIII.A

grandes-ecoles · France · centrale-maths2__mp Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts
We define the function $\psi$ (called the digamma function) on $\mathbb { R } ^ { + * }$ as the derivative of $x \mapsto \ln ( \Gamma ( x ) )$. For every real $x > 0 , \psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$. We admit that $\Gamma$ satisfies, for every real $x > 0$, the relation $\Gamma ( x + 1 ) = x \Gamma ( x )$.
Show that for every real $x > 0 , \psi ( x + 1 ) - \psi ( x ) = \frac { 1 } { x }$. 
We define the function $\psi$ (called the digamma function) on $\mathbb { R } ^ { + * }$ as the derivative of $x \mapsto \ln ( \Gamma ( x ) )$. For every real $x > 0 , \psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$. We admit that $\Gamma$ satisfies, for every real $x > 0$, the relation $\Gamma ( x + 1 ) = x \Gamma ( x )$.

Show that for every real $x > 0 , \psi ( x + 1 ) - \psi ( x ) = \frac { 1 } { x }$.
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