Abstract: The complex zeros of solutions of second order linear ODEs lie over certain curves in the complex plane called anti-Stokes lines. We consider the LiouvilleGreen (WKB) approximation for linear homogeneous second order ODEs w(z) + A(z)w(z) = 0 with A(z) meromorphic, and describe the qualitative properties of the approximate anti-Stokes lines. Based on this qualitative description, we construct a fourth order method which efficiently computes the complex zeros of solutions of second order ODEs following the path of the approximate anti-Stokes lines. We illustrate the method with the computation of the zeros of parabolic cylinder functions, Bessel functions and generalized Bessel polynomials and describe specific algorithms for the computation of these zeros.

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