It is a standard trick for evaluating difficult integrals along the real line to consider a closed-contour and "blow-up" the complex part till it vanishes, leaving us with the residues picked up along the way. This is usually done by bounding the magnitude of the integral from above with something that tends to $0$. My question is, is there an intuitive reason for why we should expect this?
My thought was that, if we project the complex plane onto a Riemann sphere, the path which appears to enlarge is actually shrunk to the point at $\infty$. I drew a (not very pretty) picture to illustrate what I mean:
In blue is a semi-circular contour in the upper-half plane, centered at the origin, and in red is the stereographic projection of this contour onto a Riemann sphere. It is clear that as the radius of the blue semi-circle $\to\infty$, the red semi-circle that corresponds to the complex portion of the contour is shrunk to the north pole of the sphere.
Is this a good way of thinking about it and is this a good enough reason for the integral along that arc to vanish (as long as it does not encounter poles on the way)?