Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 27C / a / Solution

For $\mathrm{Re}z>0$, the gamma function isRotate the integration ray through $\pi /2$, or insert a damping factor and then let it vanish. With the principal power, contour rotation givesFor the damping argument, $\int t^{\gamma -1}{e}^{-(ϵ-i)t}dt=\Gamma (\gamma )(ϵ-i{)}^{-\gamma }$, and Dirichlet tail estimates justify the limit. The condition $\gamma >0$ ensures integrability at zero; $\gamma <1$ makes the amplitude tend to zero and permits conditional convergence at infinity. At $\gamma =1$, the upper-end oscillation has no limit; larger $\gamma$ also fails ordinary improper convergence. These restrictions concern this improper integral, not its analytic regularizations.