  We produce an estimate for the $K$-Bessel function $K_{r + i t}(y)$ with
positive, real argument $y$ 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 $\mu>0$. In particular, we compute
the dominant term of the asymptotic expansion of $K_{r + i t}(y)$ as $y
\rightarrow \infty$. When $t$ and $y$ are close (or equal), we also give a
uniform estimate.
  As an application of these estimates, we give bounds on the weight-zero
(real-analytic) Eisenstein series $E_0^{(j)}(z, r+it)$ for each inequivalent
cusp $\kappa_j$ when $1/2 \leq r \leq 3/2$.
