Favorite Equation Of All Time?

[imagenius128]

What is your favorite mathematical equation/value of all time? Mine is e^{i\pi}, which equals -1.

 

[Anti-Meson]

I second that.

 

[linovari]

I've always had a soft spot for Stokes' Theorem; differential forms version:
\int_{\partial \cal C} \omega = \int_{cal C} d\omega

 

[Mensanator]

j = gmpy.divm(xyz[1]**(gen)-dp,yx,xyz[1]**(gen))//xyz[1]**(gen-ONE)

 

[MotoH]

x²-2x+3

 

[Mentallic]

d=\left|\frac{ax_1+by_1+c}{\sqrt{a^2+b^2}}\right|

- perpendicular distance between a point (x₁,y₁) and a line ax+by+c=0

I reckon the proof is so neat! Except no one else in my class appreciated it whatsoever when they learned it...

 

[Anti-Meson]

I would like to change my original thought.

May favourite is by far this:

\int_{all time}dt

 

[uart]

Ok I'll post one just for fun. :)f(x) = \frac{1}{2 \pi} \int_{-\infty}^{+\infty} \, \left\{ \int_{-\infty}^{+\infty} f(\lambda) \, e^{-i 2 \pi \omega \lambda} \, d\lambda \right\} \, e^{i 2 \pi \omega x} \, d\omega

 

[raymo39]

call me a traditionalist, but its got to be E=mc^2
its engraved on my ipod :)

 

[fourier jr]

how about functions expressed using Hankel's wacky contour:

\Gamma(z) = \frac{1}{e^{2\pi iz}-1}\int^{+\infty}_{+\infty}e^{-t}t^{z-1}dt

\zeta(s) = \frac{\Gamma(1-s)}{2\pi i}\int^{+\infty}_{+\infty}\frac{(-x)^s}{e^{x}-1}\frac{dx}{x}

 

[Algr]

1=2.

 

[Svalbard]

Mine is

All Time = 3pi/2 + 5

 

[hotvette]

y = x^x. It has a minimimum at x = 1/e.

 

[IttyBittyBit]

Why does everyone love e^i*pi = -1 so much? Because Feynman liked it? Have some originality, people :)

 

[ephedyn]

^IttyBittyBit: I didn't know that Feynman liked it - where did you read that from? Now I've more reason to like it. Coincidentally, it's the 15th of February today. He passed away exactly 22 years ago. :(

I like it for the traditional reason though, that there's e, i, pi, 0 and 1, ^, *, +, = in a single equation!

 

[torquil]

Why does everyone love e^i*pi = -1 so much? Because Feynman liked it? Have some originality, people :)

How about

<br /> 0 \neq 1<br />

Without this, maybe mathematics would not exist?

One of my lecturers in quantum field theory said that the most important (consider path integrals) equation in physics is

<br /> \log(\det(A)) =\mathrm{tr}\log(A)<br />

One of my personal favourites are

EDIT: A picture was supposed to appear here... Anyway, it was the formula that expresses e as a continued fraction. I won't bother to write it myself.

Torquil

 

[matheinste]

e^{i\pi}+1=0

This has been described as the mathematical poem, linking the sometime called big five of mathematics, e, pi, i, 0 and 1. When you consider that it involves an irrational number raised to an imaginary irrational power being equal to unity, it is, at first sight, to a non mathematician like myself, truly magical. Of course when you know a little more mathematics it is quite simple, no magic involved.

Matheinste.

 

[Phyisab****]

Ok I'll post one just for fun. :)


f(x) = \frac{1}{2 \pi} \int_{-\infty}^{+\infty} \, \left\{ \int_{-\infty}^{+\infty} f(\lambda) \, e^{-i 2 \pi \omega \lambda} \, d\lambda \right\} \, e^{i 2 \pi \omega x} \, d\omega

I'll second this one.

 

[Fightfish]

My favourite would be the time-independent Schrödinger equation

\hat{H} \psi = E\psi​

in this deceptively simple form :p

 

[Svalbard]

e^{i\pi}-1=0

This has been described as the mathematical poem, linking the sometime called big five of mathematics, e, pi, i, 0 and 1. When you consider that it involves an irrational number raised to an imaginary irrational power being equal to unity, it is, at first sight, to a non mathematician like myself, truly magical. Of course when you know a little more mathematics it is quite simple, no magic involved.

Uhm i think you got that one wrong. e^(i*pi) = -1 , not +1

 

[matheinste]

Uhm i think you got that one wrong. e^(i*pi) = -1 , not +1

Thanks for the correction.

Matheinste

 

[IttyBittyBit]

^IttyBittyBit: I didn't know that Feynman liked it - where did you read that from? Now I've more reason to like it. Coincidentally, it's the 15th of February today. He passed away exactly 22 years ago. :(

I believe it was in the 'algebra' chapter of The Feynman Lectures on Physics. That book is worth reading even for people who are not interested in physics (as is every other book by Feynman).

 

[data189]

\int_{a}^{b}f(x)dx=F(b)-F(a)

 

[Vortico]

I find these definitions of the golden ratio very elegant:

\phi = \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}
\phi = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{\ddots}}}

The proofs of these are interesting as well.

 

[Integral]

e^{i\pi}+1=0

This has been described as the mathematical poem, linking the sometime called big five of mathematics, e, pi, i, 0 and 1. When you consider that it involves an irrational number raised to an imaginary irrational power being equal to unity, it is, at first sight, to a non mathematician like myself, truly magical. Of course when you know a little more mathematics it is quite simple, no magic involved.

Matheinste.

Not only does it relate the major constants it does so by using all of the basic operations, exponentiation, multiplication, addition and equality.

 

[Mark44]

Yeah, I like this one, too.
e^{i\pi}+1=0
You raise an irrational number to a power that is an imaginary multiple of another irrational number, add 1 and get 0.

 

[Nabeshin]

G_{\mu \nu}= 8 \pi T_{\mu \nu}

Golden.

 

[Office_Shredder]

How about

<br /> 0 \neq 1<br />

Without this, maybe mathematics would not exist?

That's the opposite of an equation :p

The trace/determinant log formula is pretty cool when applied to lie algebras and lie groups (which allegedly is used in physics)

I personally like

A / ker \phi \cong I am \phi

 

[skycastlefish]

1=2