Euler's formula is a mathematical equation that establishes the deep relationship between trigonometric functions and the complex exponential function.The complex exponential is a time machine for waves. e^(iθ) = cosθ + i sinθ means that multiplying by e^(iθ) rotates a number by angle θ, combining rotation and growth.
Leonhard Euler in the 1740s experimented with series and discovered that substituting ix into the series of e^x yields two familiar series. In 1748, he published this result in 'Introductio in analysin infinitorum'. Before him, Roger Cotes had derived similar relations, but it was Euler who realized the full power of the complex exponential. Later, the formula became the key to creating complex analysis.
Euler's formula is used to represent oscillations in physics and engineering: from describing electromagnetic waves to fast Fourier transform algorithms. Its limits of applicability arise when moving to non-commutative algebras, such as quaternions, where the exponential does not decompose simply into trigonometric functions.
The most beautiful equation in mathematics: e^(iπ) + 1 = 0 brings together five important numbers — e, i, π, 1, 0 — in one equality.
| f | variable | homogeneous function of degree n — homogeneous function |
| x | variable | independent variable — variable |
| n | variable | degree of homogeneity — degree |
| f'(x) | operator | derivative of f with respect to x — derivative |
| α | variable | Scaling exponent (same as in power law) |
| K | variable | Luttinger parameter — luttinger parameter |