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Most existing algorithms for solving initial value problems in ordinary differential equations implicitly assume that all variables are real. If the real-valued assumption is removed, the solution can be extended by analytic continuation along a path of integration in the complex plane of the independent variable. This path is chosen to avoid singularities which can make the solution difficult or impossible for standard methods. We restrict our attention to Taylor series methods, although otherdoi:10.1090/s0025-5718-1980-0583495-8 fatcat:iuozl4xh75acvgmadlvtaxouzm