  Asymptotic expansions for the Bateman and Havelock functions defined
respectively by the integrals \[\frac{2}{\pi}\int_0^{\pi/2}
\!\!\!\begin{array}{c} \cos\\\sin\end{array}\!(x\tan u-\nu u)\,du\] are
obtained for large real $x$ and large order $\nu>0$ when $\nu=O(|x|)$. The
expansions are obtained by application of the method of steepest descents
combined with an inversion process to determine the coefficients. Numerical
results are presented to illustrate the accuracy of the different expansions
obtained.
