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A conjecture (revised)

The conjecture goes like this:

If the force between two points X and Y on a complex plane can be written in the form:

$F(X\rightarrow Y) = (X-Y)^{-n}$, where n is an integer,

then the average force given by a circle to Y, when Y is outside the circle, equals to the force given by the center of the circle to Y.

For n>0 it seems to hold, for n<0 I'm not yet sure. Just wonder does this belong to sth. more general (if it is correct)?

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If $X, Y$ are points in the complex plane, and I interpret $X-Y$ as the difference between complex numbers, then Cauchy's integral theorem en.wikipedia.org/wiki/Cauchy_integral_theorem should tell you that the integral around a circle not circling $Y$ to be zero. So do you perhaps mean $|X-Y|^n$ with $|\cdot|$ the Euclidean norm instead? – Willie Wong Nov 21 2010 at 0:45
If you take it to be a central force, you can write the vector of the force to be $\nabla |X-Y|^{n+1}$ instead. (Where for $n = -1$ we replace by $\log$.) Then using that derivatives commute, this requires $|X-Y|^{n+1}$ to have a mean value property. So I don't see how you can have "for $n>0$ this seems to hold". As it stands the question is not very well written: please disambiguate your notations, and even the conjecture itself. – Willie Wong Nov 21 2010 at 1:01
Sorry for the ambiguity. The first interpretation was what I wanted to mean. But Cauchy's integral theorem doesn't apply directly here, since it is NOT a line integral on the complex plane, but rather an $\phi$ integral, which is equal to an average on the circle. So the force is not centrifugal unless n=-1. (The form I wrote was $(X-Y)^{-n}$ but not $(X-Y)^{n}$, so n>0 are the normal cases where the force converges at infinity.) – Straybird Nov 22 2010 at 9:26
As I read it again, I see indeed the conjecture wasn't clearly put. I'll post it again. Thanks for the comments! – Straybird Nov 22 2010 at 9:34

closed as exact duplicate by Willie Wong, Andreas Thom, Harry Gindi, Ben Webster♦ Nov 22 2010 at 12:01

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