The asymptotic expansion of the Bateman and Havelock functions of large order and argument

Asymptotic expansions for the Bateman and Havelock functions defined respectively by the integrals [frac{2}{pi}int_0^{pi/2} !!!begin{array}{c} cossinend{array}!(xtan u- u u),du] are obtained for large real $x$ and large order $ u>0$ when $ u=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.