Asymptotics of solutions of a differential equation of second order with two turning points and a complex parameter. II

Z. A. Yanson

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 9, Tome 78 (1978), pp. 220-245

Résumé

Asymptotic formulas are constructed and rigorously justified for linearly independent solutions of a second-order differential equation with a coefficient possessing the property of finite smoothness and containing a complex parameter $\xi$ (for $\operatorname{Im}\xi=0$ the equation has two real turning points). A perturbation method is applied which consists in extending the coefficient of the equation to the complex $Z$ plane and approximating it in an $\varepsilon$-neighborhood of the real axis of this plane by a quadratic polynomial. It is proved that the leading terms of the constructed formulas expressed in terms of parabolic cylinder functions are uniform with respect to $\arg\xi$ and that the error admitted under the approximation indicated above can be estimated by the quantity $O(k^{-1/2})$, ($k\to\infty$ is the second parameter, in addition to $\xi$, on which the coefficient of the differential equation depends).