The question might a duplicate, in which case I would be more than happy if you could just redirect me to the answer.
I need to solve the following integral \begin{equation} \int_{\mathcal{C}_\rho^+} z^\delta e^{i z x} \mathrm{d} z \,, \end{equation} where $ x>0 $ real, $ \delta > 1 $ irrational and $ \mathcal{C}_\rho^+ $ is the top half of a circumference of radius $ \rho $. In other words, by explicit reparametrization $ z = \rho e^{i \varphi} $ you may rewrite the integral as \begin{equation} \int_0^\pi (\rho e^{i \varphi})^{\delta+1} e^{i \rho e^{i \varphi} x} \mathrm{d} \varphi \,. \end{equation}
The integral looks very close to the one defining an incomplete gamma function, namely \begin{equation} \gamma (\delta, y) = \int_0^y t^{\delta-1} e^{-t} \mathrm{d} t \,. \end{equation} Despite the similarity, I could not really exploit this information.
I therefore suspect that it could be solved just by "standard" means, as in deformation of the contour and residue theorem. This presents different challenges, though. On the one hand, one has to treat the multi-valued integrand carefully. On the other hand, it is not clear how to exploit the residue theorem, since there seem to be no poles besides the branching points (where it doesn't make sense to take residues, right?).
I apologize if the question is too elementary, and hope to receive useful insights from you soon!