Newton's law of gravitation in complex form

Question

In an ebook about elementary complex analysis I came across Newton's law of universal gravitation with a complex valued function in place of $r(t)$. Can somebody please explain the intuition about how a real valued function gets translated into a complex function $z(t)$? I don't understand how this was done. Thanks! (I think this might be a pretty illustrative way to connect real and complex functions, so I thought I'd put it up.)

Answer 1

The ebook considers the orbit plane

$${\mathbb{R}^2~\cong~\mathbb{C}}$$

of the body as a complex plane, and introduces complex coordinates

$$z(t)~:=~x(t)+iy(t),$$

and complex force

$$f~:=~f_x+if_y.$$

If one rewrites in real and imaginary parts, one recovers the the usual component-wise real formulation of Newton's gravitation law.