Liouville-Green expansions of exponential form, with an application to modified Bessel functions

Linear second order differential equations of the form $d^{2}w/dz^{2}-\left { {u^{2}f\left( u,z\right) +g\left( z\right) }\right} w=0$arestudied,where$\left| u\right| \rightarrow \infty $and$z$ lies in a complex bounded or unbounded domain $\mathbf{D}$. If $f(u,z)$ and $g\left(z\right)$ are meromorphic in $\mathbf{D}$, and $f(u,z)$ 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 $d^{2}w/dz^{2}-\left{ {u^{2}f\left(z\right) +g\left( z\right) }\right} w=p\left( z\right)$.