## Abstract

The methods of part III are further applied to the construction of approximations for the fundamental solution and base functions of part II in terms of higher transcendental functions. The domain of validity is now the complete half-strip {z; 0 $\leq $ Re z $\leq \frac{1}{2}\pi $, Im z $\geq $ 0} without exceptional point. Relative remainder estimates are again uniformly valid provided they are bounded. Specifically, approximations are obtained in terms of: (a) Airy functions, applicable if $\lambda \neq \pm $ 2h$^{2}$; (b) parabolic cylinder functions, applicable if $|\lambda|\leq $ 4h$^{2}$, including $\lambda $ = $\pm $ 2h$^{2}$; (c) Bessel functions, applicable if $|\lambda|\geq $ 4h$^{2}$; these formulae have maximum relative error $\lambda ^{-\frac{3}{2}}$h$^{2}$O(1) on the half-strip, even if h is arbitrarily small, provided only that $\lambda ^{-1}$ is bounded. This is significantly better when $\lambda $/h$^{2}$ is large than the corresponding estimate, $\lambda ^{-\frac{1}{2}}$O(1), for the Airy function approximations. Certain more refined estimates for the auxiliary parameters introduced in part II are also obtained.