Advantages of complex time in Classical Mechanics

Let us consider the problem in Newtonian mechanics: $m\ddot x(t)=F(x(t))$, with $F$ known. Let us promote time to a complex variable, and admit the solution $x(t)$ (1) can be analytically continued to some finite region of the complex plane which contains an interval of the real axis $I=(t_1, t_2)\subset $ of the region enclosed by the simple curve $C$ and (2) it is analytic in this region. Therefore, we can now apply a complex analysis identity to write derivatives as integrals, obtaining:

$\frac{m}{2\pi i}\oint_C\frac{x(t')}{(t'-t)^3}dt'=F(x(t))$

To obtain the solution to Newton's equation in real time, we may simply take $t\in I$ in this expression.

At this point, I have 2 questions:

1. Could this approach be of any advantage, in any case, to the usual approach of simply solving the ODE?

2. If the answer to (1) is no, could it be of any advantage to study the analytic continuation of the potential, the force or the very solution to the complex plane? For instance, when working with dispersion relations in scattering theory, knowing the analytic structure of the S matrix is enough to formulate dispersion relations which we can then use (cf. Roy equations, forward dispersion relations, DKPY, etc.) even if we never solve the theory