What is your favorite proof?

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z4lis
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Joined: Mon Mar 03, 2008 10:59 pm UTC

Re: What is your favorite proof?

Postby z4lis » Mon Apr 04, 2011 5:03 am UTC

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. :P
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.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

dissonant
Posts: 63
Joined: Sat Jan 24, 2009 10:33 am UTC

Re: What is your favorite proof?

Postby dissonant » Mon Apr 04, 2011 9:09 am UTC

Last Thursday I witnessed a proof of Urysohn's lemma. It was exceptionally beautiful. My professor spent about five minutes drawing contour lines to illustrate the separation of the topological space using dyadic fractions.

Also, there is a proof of the Borsuk-Ulam Theorem in the case of the 1-sphere which proceeds via the intermediate value theorem which is also very pretty. I can not decide between the two; they are both my favourite at the moment.


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