The O.D.E. studied is of second order of the Schroedinger type constaining a small parameter, considered as the Planck constant, and it has a turning point and a regular singular point both situated at the origin. A case where the turning point and the singular point coincide has a very typical figure of the existence region (canonical domain) of the asymptotic solutions. Several cases of different order of the turning point are shown.
"On Asymptotics of a Second Order Linear ODE with a Turning-regular Singular Point." Tokyo J. Math. 33 (1) 145 - 157, June 2010. https://doi.org/10.3836/tjm/1279719583