Laplace transform of a power function

Example

Let $\alpha > -1$ be real and $f(t) = t^\alpha$ for $t > 0$. Its Laplace transform is $$\mathcal{L}\{t^\alpha\}(s) = \frac{\Gamma(\alpha + 1)}{s^{\alpha + 1}} \quad (\operatorname{Re} s > 0),$$ where $\Gamma$ is the gamma function and $s^{\alpha + 1}$ is the principal power. For $\alpha = n \in \N$ this is $\mathcal{L}\{t^n\}(s) = n!/s^{n + 1}$, with $n!$ the factorial. Check for real $s > 0$: the substitution $u = st$ gives $$\int_0^\infty t^\alpha e^{-st}\, dt = \frac{1}{s^{\alpha + 1}} \int_0^\infty u^\alpha e^{-u}\, du = \frac{\Gamma(\alpha + 1)}{s^{\alpha + 1}}.$$ The integral converges at $0$ because $\alpha > -1$, and at $\infty$ because of the factor $e^{-st}$. Both sides are holomorphic in $s$ on $\operatorname{Re} s > 0$ and agree for real $s > 0$, so they agree on the whole half-plane. For example, $\mathcal{L}\{t^{-1/2}\}(s) = \sqrt{\pi / s}$, since $\Gamma(1/2) = \sqrt{\pi}$.