Equations

This page is devoted to cool or interesting mathematical 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!

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