= Solution
For $\operatorname{Re}z>0$, the <gamma function> is
$$
\Gamma(z)=\int_0^\infty t^{z-1}e^{-t}dt.
$$
Rotate the integration ray through $\pi/2$, or insert a damping factor and then let it vanish. With the principal power, <contour rotation> gives
$$
\boxed{\int_0^\infty t^{\gamma-1}e^{it}dt=e^{i\pi\gamma/2}\Gamma(\gamma),\qquad0<\gamma<1}.
$$
For the damping argument, $\int t^{\gamma-1}e^{-(\epsilon-i)t}dt=\Gamma(\gamma)(\epsilon-i)^{-\gamma}$, 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.
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