Throughout the rest of the problem, we denote $\Gamma$ the function defined on $\mathbb { R } ^ { + * }$ by $\Gamma ( x ) = \int _ { 0 } ^ { + \infty } t ^ { x - 1 } e ^ { - t } \mathrm {~d} t$. We admit that $\Gamma$ is of class $\mathcal { C } ^ { \infty }$ on its domain of definition, takes strictly positive values and satisfies, for every real $x > 0$, the relation $\Gamma ( x + 1 ) = x \Gamma ( x )$. Let $x$ and $\alpha$ be two strictly positive real numbers. Justify the existence of $\int _ { 0 } ^ { + \infty } t ^ { x - 1 } e ^ { - \alpha t } \mathrm {~d} t$ and give its value as a function of $\Gamma ( x )$ and $\alpha ^ { x }$.
Throughout the rest of the problem, we denote $\Gamma$ the function defined on $\mathbb { R } ^ { + * }$ by $\Gamma ( x ) = \int _ { 0 } ^ { + \infty } t ^ { x - 1 } e ^ { - t } \mathrm {~d} t$. We admit that $\Gamma$ is of class $\mathcal { C } ^ { \infty }$ on its domain of definition, takes strictly positive values and satisfies, for every real $x > 0$, the relation $\Gamma ( x + 1 ) = x \Gamma ( x )$.
Let $x$ and $\alpha$ be two strictly positive real numbers. Justify the existence of $\int _ { 0 } ^ { + \infty } t ^ { x - 1 } e ^ { - \alpha t } \mathrm {~d} t$ and give its value as a function of $\Gamma ( x )$ and $\alpha ^ { x }$.