Most Beautiful Math Equations

February 7, 2008 — Karthik

This is a (subset of the) list of the most beautiful math equations ever derived, according to Ed Pegg Jr and Eric Weisstein (Mathworld). You can check the whole list here.

Archimedes’ Recurrence Formula
Let a_n and b_n be the perimeters of the circumscribed and inscribed n-gon and a_{2n} and b_{2n} the perimeters of the circumscribed and inscribed 2n-gon. Then
a_{2n}=\frac{2 a_n b_n}{a_n+b_n}, \quad b_{2n}=\sqrt{a_{2n}b_n}, \quad a_\infty=b_\infty
Successive application gives the Archimedes algorithm, which can be used to provide successive approximations to \pi.

Euler Formula
e^{i\pi}+1=0
This equation connects the fundamental numbers i, \pi, e, 1, and 0, the fundamental operations +, \times, and exponentiation, the most important relation =, and nothing else. Gauss is reported to have commented that if this formula was not immediately obvious, the reader would never be a first-class mathematician.

Euler-Mascheroni Constant
The Euler-Mascheroni constant gamma, sometimes also called `Euler’s constant’ or `the Euler constant’, is defined as the limit of the sequence
\gamma = \lim_{n \rightarrow \infty}\left(\sum_{k=1}^n \frac{1}{k}-\ln n\right)

\gamma has the numerical value 0.577215664901532860606512090082402431042…

Riemann Hypothesis
The Riemann hypothesis is a deep mathematical conjecture which states that the non-trivial Riemann zeta function zeros, i.e., the values of s other than -2, -4, -6, \ldots such that \zeta(s)=0 (where \zeta(s) is the Riemann zeta function) all lie on the “critical line” \sigma=\mathbb{R}[s]=1/2 (where \mathbb R[s] denotes the real part of s).

The Riemann zeta-function \zeta(s) is the function of a complex variable s initially defined by the following infinite series:

\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}

The Riemann hypothesis can be stated as:
\zeta(\alpha+i\beta)=0 and \beta \neq 0 implies \alpha=\frac{1}{2}.

Gaussian Integral
The Gaussian integral, also called the probability integral is the integral of the one-dimensional Gaussian function over (-\infty,\infty)

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

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