Consider the following integral
$I = \int_{-\sqrt{b^{2} + a^{2}}}^{\sqrt{b^{2}+a^{2}}} (a^{2} - z^{2})^{\lambda}e^{- i \omega z}\mathrm{d}z$
In this integral, $a$, $b$ and $\omega$ are real numbers. The exponent $\lambda$ is also a real number, but can be taken to be a half-integer for simplicity.
My question is regarding the presence of branch points on the region of integration. Note that the branch points are at $z = \pm a$ (one can choose $a$ to be positive for simplicity). How does one proceed with this integration? More specifically, what would be the correct choice of contour?
Thanks in advance!