The integration is
$$\int_{c-i\infty}^{c+i\infty} dz\,\frac{e^{a\sqrt{z+m^2}+bz}}{\sqrt{z+m^2}-m}$$
I understand there is a branch cut at $z = -m^2$. But a little confused about the behavior at $z = 0$. If we put $z=0$, we get a pole. Is that a branch cut too? In this case, how many branch cuts are there? And branch cut runs from 0 to infinity or $-m^2$ to infinity?