Your favorite equation

My favorite equation is
$$\int_{\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}.$$
What's yours??

Note: This is a real questions and I expect a response from everyone in the class. The point is to simply enter something that uses LaTeX to typeset some mathematics. You should use double dollar signs to place your formula into display mode and it should involve a non-trivial degree of typesetting - i.e., it needs an integral or a sum and a fraction or a root and a Greek letter or a funky symbol or...

And it'd better be cool!

And preferably mathematically correct!!

My favorite equation is,

$$e^{ix}=\cos{x}+i\sin{x}$$

Wooooo

Here's mine:

$$-\frac{GMm}{r^2}=\frac{M}{M+m}m\ddot{r}$$

My favorite is........

$$a^{p-1} \equiv 1\ (mod\ p)$$

WOOOOOO!!!!

My favorite equation is the closed form of the Fibonacci sequence.

Def: The Fibonacci numbers are the numbers $F_n=F_{n-1}+F_{n-2}$ given $F_0=0$ and $F_1=1$.

Thm: The $n^{th}$ Fibonacci number is given by the closed formula for any $n \in \mathbb{N}$:
$$F_n=\frac{1}{\sqrt{5}} \left( \frac{1+\sqrt{5}}{2} \right)^n - \frac{1}{\sqrt{5}} \left( \frac{1-\sqrt{5}}{2} \right)^n$$

Proof: Available here.

I know I'm going a little basic, but I just love the quadratic formula solving for x in $ax^2+bx+c=0$:
$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

I'll go with the Taylor Series equation:

$${f(x)} = \sum_{n=0}^{\infty} {\frac{f^{n}(x_0)}{n!}}{(x-x_0)}^n$$

Wooo.

I know this is basically the same equation that Mark showed us in class but,
$$e^{i\pi}+1=0$$
is my favorite equation. It has been since Mark showed it to us in Calc 2 spring of 2013. it is my favorite because it combines so many mathematical operations in it.

There is addition/subtraction, multiplication/division, imaginary numbers, exponents, and Trig all in one small equation.

My favorite equation is the analytical continuation of the Factorial.

$$ n! = \int_{0}^{\infty}x^{n}e^{-x}dx$$

This is cool!!!

Newton's method for finding roots!
$$n(x) = x - \frac{f(x)}{f'(x)}$$

I could lie and say this is my favorite equation, but I honestly chose an equation that would challenge by LaTeX skills (not hard to do!) $$\prod_{k=1}^{n-1}\sin\left(\frac{k\pi}{n}\right)=\frac{n}{2^{n-1}}$$

"Eisenstein's lemma", which states that for distinct odd primes p, q, the value of the Jacobi symbol, denoted, $\left(\frac qp\right)$, is:

$$\left(\frac qp\right) = (-1)^{\sum_u \left \lfloor qu/p \right \rfloor}.$$

My favorite equation is the mass of all self replicating Bender units from Futurama after $n$ phases of reproduction.

$$M=\sum_{n=0}^{\infty} 2^{n}\bigg[\frac{M_{0}}{2^{n} (n+1)}\bigg]$$

My favorite equation right now is Einstein's equation,

$$R_{\mu\nu} - \frac{1}{2}R g_{\mu\nu} = 8 \pi G T_{\mu\nu}.$$

I'm pretty fond of Tupper's Self-Referential Formula:

$$\frac{1}{2} < \lfloor \text{mod} (\lfloor \frac{y}{17} \rfloor 2^{-17\lfloor x\rfloor - \text{mod}(\lfloor y\rfloor,17)},2)\rfloor$$

It graphs itself, fun fact. It's a cool mix of computer science and math.

$$K(f)-K(i)=\int_{s_0}^{s_f} F(s)ds$$

Work Energy Theorem.

Wish I could like it twice!

Good news everyone! This is brilliant.

The Navier-Stokes Equation holds a special place in my heart...

$$\rho\bigg(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\bigg) = -\nabla p + \nabla\cdot \mathbf{T} + \mathbf{f}$$

Look at that beautiful equation.

For my favorite equation you will need to know the following:

1. $a,b\in \mathbb{Z}$ with $(a,b)=1$
2. $D$ is a non-perfect-square integer
3. $frac{a}{b}$ is assumed to be equal to $\sqrt{D}$
4. $\lambda<\sqrt{D}<\lambda +1$ and
5. The equation is part of a proof (by contradiction) for the irrationality of a non-perfect-square integer by Richard Dedekind.

Now here is my favorite equation:

$$\frac{Db-\lambda a}{a-\lambda b}=\frac{a}{b}$$

Dedekind shows that $Db-\lambda a < a$ and $a-\lambda b < b$ which shows that $\frac{a}{b}$ can be reduced which gives the contradiction.
But here's the coolest part! You can take this algebraic proof and come up with a geometric proof that follows the algebra perfectly! Well, at least for the $\sqrt{2}$, $\sqrt{3}$, $\sqrt{6}$, $\sqrt{8}$, and the $\sqrt{12}$...

Here are their pictures.