Talk:Euler's Formula

I vote we remove the boxes around the formulas on this page. Who's with me? --Cynic 14:17, 27 April 2008 (UTC)

I'm for that --Joe 14:20, 27 April 2008 (UTC)

Looks better, I think. Also, I think the derivative of the quotient expression ought to be illustrated as well.--MathMonkeyMan 20:51, 27 April 2008 (UTC)

Was the last box left in intentionally? It actually might be a good idea to box the final step in all the proofs so it's clear when you have reached the end and you can see what was being proved. Or add a "Theorem:" line to the beginning of each proof.--Cynic 21:28, 27 April 2008 (UTC)

I think the boxes are a good idea, I vote we leave them.

Euler's formula can be taken as a definition rather than something one can prove: the exponential is defined for real numbers, and one wants to extend it to complex numbers in some way. You can prove that the expression must be that, but one should clearly state the assumptions; for example, that the only holomorphic extension of the exponential to the complex plane must be given by that expression, or that the only continuous extension is given by that expression. These are different results, and here it is not clear what the assumptions are. When you want to prove that $$e^{i\theta}$$ has a certain expression, what is your definition of $$e^{i\theta}$$?--Cañizo 12:34, 19 February 2009 (UTC)

I think that when this page was written, the site was still young. It was only a month or two ago that real analysis was tackled properly, and until you've defined some basic stuff from calculus, you can't really define $$e$$, let alone $$e^{\imath \theta}$$.

Once I've got some of the boring topology out of the way (nearly as boring as that real analysis stuff which is so utterly tedious it makes my teeth itch) I'm going to be in a position to take on complex analysis a bit more seriously than I have done up till now, and I hope to bring it into line with the other stuff. Yes I know you don't rate my work on analysis much, no nor do I, I'm taking a step back before I think about how best to rationalise the pages on continuity, convergence and limits. Do we "just" give a topological definition and blandly use the fact that $$\R$$ and $$\C$$ are topological spaces? Or do we want to provide proofs, definitions and the like for all stages of mathematical understanding? My view is the latter. --Matt Westwood 21:36, 19 February 2009 (UTC)