grandes-ecoles 2015 Q1a

grandes-ecoles · France · x-ens-maths2__mp Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts

Show that $\Gamma$ is well defined and that for all $y > 0 , y \Gamma ( y ) = \Gamma ( y + 1 )$. Deduce that, for all $n \in \mathbb { N } , \Gamma ( n + 1 ) = n !$.
Recall that $\Gamma : ] 0 , + \infty [ \rightarrow \mathbb { R }$ is defined by $\Gamma ( y ) = \int _ { 0 } ^ { \infty } e ^ { - t } t ^ { y - 1 } d t$ and that $\Gamma \left( \frac { 1 } { 2 } \right) = \sqrt { \pi }$.

Show that $\Gamma$ is well defined and that for all $y > 0 , y \Gamma ( y ) = \Gamma ( y + 1 )$. Deduce that, for all $n \in \mathbb { N } , \Gamma ( n + 1 ) = n !$.

Recall that $\Gamma : ] 0 , + \infty [ \rightarrow \mathbb { R }$ is defined by $\Gamma ( y ) = \int _ { 0 } ^ { \infty } e ^ { - t } t ^ { y - 1 } d t$ and that $\Gamma \left( \frac { 1 } { 2 } \right) = \sqrt { \pi }$.

Paper Questions

Q1a Q1b Q2a Q2b Q2c Q2d Q3a Q3b Q3c Q3d Q3e Q4 Q5 Q6a Q6b Q6c Q6d Q7a Q7b Q8 Q9 Q10a Q10b Q11 Q12 Q13 Q14a Q14b Q15a Q15b Q15c