Universal Law of Gravitation described by complex functions

[coolbeans777]

Main Question or Discussion Point

I didn't put this in any of the physics sections, because I'm only learning this as it pertains to complex analysis. This was just an example provided in a free complex analysis textbook. Basically what I'm asking is why for the law of gravitation, f = GMm/r(t)^2, and it equals the equation in the picture when described by complex functions.

http://i.minus.com/iblXBR9aafGTaZ.png [Broken]

 

[Mute]

It looks to me like they're just using the complex function z(t) = x(t) + iy(t) in place of a vector function r(t) = (x(t),y(t)). Since the only information you really care about is x(t) and y(t), the two representations will give you the same results.

 

[HallsofIvy]

Yes. This only uses the fact that the complex plane has two real dimensions. It has nothing to do with the algebraic properties that distinguish the complex plane from R²