Leonhard Euler

1. Euler’s Identity

Published in the 1740s, this formula became iconic for uniting analysis, geometry, and arithmetic at a time when complex numbers were still controversial.

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

2. Euler’s Formula (Complex Exponentials)

Euler introduced this in the mid-18th century, giving complex numbers a geometric interpretation that transformed trigonometry and analysis.

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

3. Euler’s Polyhedron Formula

Discovered in 1752, this relation marked one of the earliest results in topology, long before the field had a name.

$$ V - E + F = 2 $$

4. Basel Problem Solution

Euler solved this famous problem in 1734, astonishing contemporaries by linking infinite series to \(\pi\).

$$ \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} $$

5. Euler–Lagrange Equation (Euler’s Original Form)

Euler developed this equation in the 1740s while founding the calculus of variations, later refined with Lagrange.

$$ \frac{d}{dx}\left(\frac{\partial L}{\partial y'}\right) - \frac{\partial L}{\partial y} = 0 $$

6. Euler’s Totient Formula

Introduced in the 1760s, this identity emerged from Euler’s systematic generalization of Fermat’s number-theoretic ideas.

$$ \sum_{d \mid n} \varphi(d) = n $$

7. Euler’s Product Formula for the Zeta Function

Euler discovered this in 1737, revealing for the first time a deep analytic link between prime numbers and infinite series.

$$ \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}} $$

8. Euler’s Identity for Homogeneous Functions

Formulated in the 1750s, this result arose from Euler’s broader effort to formalize scaling laws in mechanics and geometry.

$$ x f_x + y f_y = k f $$