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 $e^{i \theta}$.

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)

Don't take me wrong, I just say this to make the page better... For the moment I'm enthusiastic about it, and find it real fun to edit things, then find out that other people edited them, try to discuss about this or that... I can disagree on how to best write things, but that's the point! A lot of disagreement may converge to something nice. Of course this is just starting, and the questions you mention have to be taken one by one.--Cañizo 00:16, 21 February 2009 (UTC)

Added a "proof" which works more as a definition of the (complex) logarithm function than a proof itself. It's wordy and long because it appeals to intuition and common insights about complex numbers.--Misael.G.Mx 00:01, 6 December 2011 (CST)
 * It certainly opens up a few avenues which we haven't got round to doing yet (I was going to get round to doing complex analysis once I'd covered the appropriate topological background but never got back to it). Mind, it needs a lot of tidying up and serious restructuring. "Wordy and long" is not the usual style that ProofWiki's philosophy is based on. It's got definitions, lemmas, more lemmas, subproofs, who knows what-all, and all of those really belong on their own pages. Lemmas are often useful somewhere else not just in the proof they are raised. And there's some of colloquial language used grammatically inaccurately, which makes it look unprofessional. I'm also not too sure about circularity. However, I'm not in the mood for going over it at the moment, I'm doing other stuff and (my usual complaint) I'm seriously mentally exhausted by the day job at the moment. If anyone else wants to take this one on, feel free. --prime mover 15:19, 6 December 2011 (CST)