I am a physics student. In calculating transition amplitude for Klein-Gordon real-scalar field, I encountered the integral,
$$ \frac{-i}{2(2\pi)^2\Delta x} \int^{\infty}_{-\infty} \,dk \frac{ke^{ik\Delta x}}{\sqrt{k^2+m^2}} $$
I can see here the integrand has branch cuts at $ k= \pm im $ However, later they do a change of variables $ z= -ik $ and then the integral becomes,
$$ \frac{1}{2(2\pi)^2\Delta x} \int^{\infty}_{m} \,dz \frac{ze^{-z\Delta x}}{\sqrt{z^2-m^2}} $$
And it is said that they can wrap the contour around the upper branch cut for $ \Delta x > 0 $
I am not able to see how this transformation happens and how the contour can be wrapped around the upper branch. Thanks for your inputs.