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Using the saddle-point method, we compute an asymptotic, as $y \rightarrow\infty$, for the $K$-Bessel function $K_{r + i t}(y)$ with positive, realargument $y$ and of large complex order $r+it$ where $r$ is bounded and $t = y\sin \theta$ for a fixed parameter $0\leq \theta\leq \pi/2$ or $t= y \cosh \mu$for a fixed parameter $\mu>0$. Our method gives an illustrative proof, usingelementary tools, of this known result and explains how these asymptotics comeabout.  As part of our proof, we prove a new result, namely a novel integralrepresentation for $K_{r + i t}(y)$ in the case $t= y \cosh \mu$. This integralrepresentation involves only one saddle point.

An asymptotic for the K-Bessel function using the saddle-point method