Consider the function

$$ f(\vec{x}) = \int_{\Bbb R^3} {\frac{ e^{-i\,\vec{x}\cdot\vec{k}}}{\sqrt{\vec{k}^2 + m^2}}} d^3 k $$

from Zee's *Quantum Field Theory in a Nutshell*. He argues like this: “the square root cut starting at $±im$ tells us that the characteristic value of $|\vec{k}|$ in the integral is of order $m$, leading to an exponential decay $\sim e^{−m|\vec{x}|}$”. I cannot understand what he means. It would be nice if someone can expand or suggest a reference for me. Thanks.