Most Beautiful Equation?

[Demosthenes]

e^{i \pi}+1 = 0

[WetWired]

Interesting. I don't think that was covered in any math class I've taken.

[Demosthenes]

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

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

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

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."

[Knight Sir Rick]

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:

\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):

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

Then there's the Mandelbrot set, given by the quadratic recurrence equation (http://mathworld.wolfram.com/QuadraticRecurrenceEquation.html)

z_{n+1} = z_n^2 + C

Interestingly, it's possible to use a very similar, cubic equation to produce a three-dimensional mandelbulb (http://www.skytopia.com/project/fractal/mandelbulb.html). (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.

[Demosthenes]

Nice post! All great choices.

[D3V]

ITT: copy and pasting theories by laureates who literally spend their lifetime formulating nonsense and circlejerk with fellows into eternity

[Asamin]

Nice post! All great choices.

Imma go ahead and not copy-paste so d3v can't rage but I would say it's the infinite sum of n

Most of the solution is assumptions based on infinite series (the controversial part is that whatever theorem they taught us that if a sum cycles between two points, it's convergence is the average of those two points) solves to -1/12. So theoretically, if you add up every natural positive number into infinity, you get -1/12.

Wiki if curious. I found this originally from a numberphile video that I'll also link where they do the proof.
Wiki (https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF)
Numberphile video (https://www.youtube.com/watch?v=w-I6XTVZXww)

[Chruser]

Imma go ahead and not copy-paste so d3v can't rage but I would say it's the infinite sum of n

Most of the solution is assumptions based on infinite series (the controversial part is that whatever theorem they taught us that if a sum cycles between two points, it's convergence is the average of those two points) solves to -1/12. So theoretically, if you add up every natural positive number into infinity, you get -1/12.

Wiki if curious. I found this originally from a numberphile video that I'll also link where they do the proof.
Wiki (https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF)
Numberphile video (https://www.youtube.com/watch?v=w-I6XTVZXww)


Zeta function regularization is fake and gay.

YuIIjLr6vUA

In all seriousness, even though I dislike the 1 + 2 + 3 + ... = -1/12 "identity" that arises from analytic continuation of the Dirichlet series \sum_{n=1}^\infty\frac{1}{n^s} into the Riemann zeta function \zeta(s), the function itself is fantastic. Also, Euler's connection between the Dirichlet series (and thus the Riemann zeta function) and the prime numbers is possibly my favorite equation:

\sum_{n=1}^\infty\frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}}

You can do all sorts of fun things with it. For instance, let s=1. Then the left-hand side is the harmonic series (https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)), which diverges. Thus the right-hand side must also diverge, which can only happen if there is an infinite number of primes. Euclid's proof (https://en.wikipedia.org/wiki/Euclid%27s_theorem#Euclid%27s_proof) was rubbish in comparison.