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 Most Beautiful Equation?
 Posted 2010-03-30, 12:08 AM $\reverse e^{i \pi}+1 = 0$
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Demosthenes

 Posted 2010-03-30, 07:16 AM in reply to Demosthenes's post "Most Beautiful Equation?" Interesting. I don't think that was covered in any math class I've taken.
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 WetWired

Posted 2010-03-30, 10:32 AM in reply to WetWired's post starting "Interesting. I don't think that was..."
WetWired said: [Goto]
 Interesting. I don't think that was covered in any math class I've taken.
I do think it is the most beautiful equation I've ever seen. To me, if there is any evidence for a God in this universe, that is it.

It stems from

$\reverse e^{ix}=Cos(x)+iSin(x)$.

This can be seen from a Taylor expansion about 0 of e^x, Cos(x) and Sin(x). It is very useful in complex variable theory, as one can now say

$\reverse A+Bi = \sqrt{ \|A^2\| + \|B^2\|}e^{i \theta}$

where theta is the angle a line from the origin to A+Bi makes with the positive real axis on the complex plane. Furthermore, if theta is made a variable that varies from 0 to 2*pi then

$\reverse e^{i \theta}$

is the unit circle.

I only created this thread with the intention of checking the functionality of LaTeX. But perhaps it is a subject that merits some discussion. Could someone move it to a more appropriate section, and title it "Most Beautiful Equation."

Last edited by Demosthenes; 2010-03-30 at 11:03 AM.
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Demosthenes

Posted 2010-03-31, 05:52 AM in reply to Demosthenes's post starting "I do think it is the most beautiful..."
Demosthenes said: [Goto]
 I do think it is the most beautiful equation I've ever seen. To me, if there is any evidence for a God in this universe, that is it.

Good choice. Feynman thought it to be "the most remarkable formula in mathematics".

Euler's solution to the Basel problem (which inspired Riemann when he defined his zeta function) has an elegant result:

$\reverse \sum_{n=1}^\infty {1 \over n^2} = \lim_{n \to \infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{n^2}\right) = \frac{\pi ^2}{6}$

The Gaussian integral expresses a beautiful result, too (which can be evaluated using polar coordinates):

$\reverse \int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$

Then there's the Mandelbrot set, given by the quadratic recurrence equation

$\reverse z_{n+1} = z_n^2 + C$

Interestingly, it's possible to use a very similar, cubic equation to produce a three-dimensional mandelbulb. (Although as far as I know, it hasn't yet been shown so far that this is a "true" 3d analogy of the mandelbrot set.)

More later. Also, as the public speaker of ancient Greece noted, LaTeX isn't working properly on this site. Solve.
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Knight Sir Rick

 Posted 2010-03-31, 07:56 PM in reply to Knight Sir Rick's post starting "Good choice. Feynman thought it to be..." Nice post! All great choices.
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Demosthenes

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