  Linear second order differential equations of the form $d^{2}w/dz^{2}-\left
\{ {u^{2}f\left( u,z\right) +g\left( z\right) }\right\} w=0$ are studied, where
$\left| u\right| \rightarrow \infty $ and $z$ lies in a complex bounded or
unbounded domain $\mathbf{D}$. If $f\left( u,z\right) $ and $g\left( z\right) $
are meromorphic in $\mathbf{D}$, and $f\left(u,z\right) $ has no zeros, the
classical Liouville-Green/WKBJ approximation provides asymptotic expansions
involving the exponential function. The coefficients in these expansions either
multiply the exponential, or in an alternative form appear in the exponent. The
latter case has applications to the simplification of turning point expansions
as well as certain quantum mechanic problems, and new computable error bounds
are derived. It is shown how these bounds can be sharpened to provide realistic
error estimates, and this is illustrated by an application to modified Bessel
functions of complex argument and large positive order. Explicit computable
error bounds are also derived for asymptotic expansions for particular
solutions of the nonhomogeneous equations of the form $d^{2}w/dz^{2}-\left\{
{u^{2}f\left(z\right) +g\left( z\right) }\right\} w=p\left( z\right)$.
