Yakk wrote:Mike_Bson wrote:My favorite is definitely using differential equation to prove Euler's Formula.

If f(x) = e^(ix), then f'(x) = i*e^(ix) = i*f(x). If f(x) = cos(x) + isin(x), then f'(x) = -sin(x) + icos(x) = i*f(x). So these both are solutions to the differential equation y' - iy = 0, and they both satisfy f(0) = 1. Thus, e^(ix) = cos(x) + isin(x).

What is the theorem that shows that this has to be unique?

I'll take a wild stab in the dark and say Banach's Fixed Point Theorem. You turn the differential equation into an integral equation, then make that integral equation an operator on a function space, and then make the interval over which you integrate small enough to apply the theorem. To get the solution for all time, you do an iterative method, taking old data and plugging it in again to keep the solution going. I don't remember all the details, though. I'm also afraid of the complex plane.

EDIT: Unless you already knew that, and were just pointing out there's much more to it.

EDIT2: My fear of complex numbers is justified, since I'm not sure how to make this into an integral equation. Ignore me. :\

EDIT3: You could possibly still do what I suggested, using path integrals. I need to stop doing edits.