## An equation of Ince: The confluent Mathieu equation.(English)Zbl 0761.34005

The author studies the second order linear differential equation of the form $$xY''+{1\over 2}Y'+(A-B^ 2x-Kx^ 2)Y=0$$ which has two singularities: an elementary singularity at the origin and an irregular singularity of the third type at infinity. He exhibits an explicit solution of this equation in the form of power series and investigates asymptotic properties of solutions at infinity.
Reviewer: L.Janos (Praha)

### MSC:

 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. 34A05 Explicit solutions, first integrals of ordinary differential equations 33E10 Lamé, Mathieu, and spheroidal wave functions 34M99 Ordinary differential equations in the complex domain 34D05 Asymptotic properties of solutions to ordinary differential equations 34A30 Linear ordinary differential equations and systems 34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.) 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations