Asymptotics of solutions of second-order differential equations with two turning points and a complex parameter. I

Asymptotic representations are obtained for the solutions of a second-order linear differential equation with coefficient having finite smoothness and containing a complex parameter $\zeta$. The asymptotic solutions are expressed in terms of parabolic cylinder functions, and the estimate for the correction to the leading term of the asymptotic expression is uniform with respect to $\arg\zeta$.