Euler's formula states that
- eix = cos x + i sin x;
for any real number x. Here, e is the base of the natural logarithm, i is the imaginary unit (see complex numbers) and sin and cos are trigonometric functions. The complex number cos x + i sin x is oftentimes refered to as cis x for short.
This formula can be interpreted as saying that the function eix traces out the unit circle in the complex number plane as x ranges through the real numbers. Here, x is the angle that a line connecting the origin with a point on the unit circle (see vectors) makes with the positive real axis, measured counter clockwise and in radians.
The proof is based on the Taylor series expansions of the exponential function ez (where z is a complex number) and of sin x and cos x for real numbers x. In fact, the same proof shows that Euler's formula is even valid for all complex numbers x.
Euler's formula was proved (in an obscured form) for the first time by Roger Cotes in 1714, then rediscovered and popularized by Euler in 1748. It is interesting to note that neither of these men saw the geometrical interpretation alluded to above: the view of complex numbers as points in the plane arose only some 50 years later.
The formula provides a powerful connection between analysis and trigonometry. It is used to represent complex numbers in polar coordinates and allows the definition of the logarithm for complex arguments. By using the exponential laws ea+b = ea · eb and (ea)b = eab (which are valid for all complex numbers a and b), one can also readily derive several trigonometric identities as well as de Moivre's formula from it. Euler's formula allows one to interpret the sine and cosine functions as mere variations of the exponential function:
- cos x = (eix + e-ix) / 2
- sin x = (eix - e-ix) / (2i)
These formulas can even serve as the definition of the trigonometric functions for complex arguments x.
In differential equations, the function eix is often used to simplify derivations, even if the final answer is a real function involving sine and cosine. The most remarkable formula in the world is an easy consequence of Euler's formula.
In electrical engineering and other fields, signals that vary periodically over time are often described as a combination of sine and cosine functions (see Fourier analysis), and these are more conveniently expressed as exponential functions with imaginary exponents, using Euler's formula.