Fundamental solutions which decay exponentially at $\infty $ together with their first derivatives are called principal fundamental solutions. Such solutions of elliptic equations in $R^m $, $m \geqq 2$, play an important role in the study of pseudo-parabolic equations in $R^m \times R$. We establish the behavior as $| {x - y} | \to \infty $ and as $| {x - y} | \to 0$ of a function H defined for x, $y \in R^m $, $x \ne y$, which is basic for the construction of the principal fundamental solution G. It is known that $G - H = O(H)$. Since H is closely related to Bessel functions, complex analysis is used to determine its behavior.