# Equations

The first 10 are from a post in my real-life blog, from a post called “10 Mind Blowing Mathematical Equations.”

1. Euler’s Identity

$e^{i\pi} + 1 = 0$

2. The Euler Product Formula

$\displaystyle \sum_{n} \frac{1}{n^s} = \prod_{p} {\frac{1}{1 - \frac{1}{p^s}}}$

3. The Gaussian Integral

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

4. The Cardinality of the Continuum

${\mathbb{R}} \sim {2^{\mathbb{N}}}$

5. The Analytic Continuation of the Factorial

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

6. The Pythagorean Theorem

$a^2 + b^2 = c^2$

7. The Explicit Formula for the Fibonacci Sequence

$F(n) = \frac{(\varphi)^n - (-\frac{1}{\varphi})^n}{\sqrt{5}}$

8. The Basel Problem

$\displaystyle1 + \frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\cdots =\frac{\pi^2}{6}$

9. The Harmonic Series

$\displaystyle1 + \frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots =\infty$

10. The Explicit Formula for the Prime Counting Function

$\displaystyle\pi(x) = \sum_{n = 1}^\infty \frac{\mu(n)}{n} J(\sqrt[n]{x})$

where ${J(x)}$ is defined as

$\displaystyle J(x) = Li(x) + \sum_{\rho} Li(x^\rho) - \log 2 + \int_{x}^\infty \frac{dt}{t(t^2 - 1)\log t}$

More to come!