### Euler's formula

(formula relating exponentials to trigonometric functions)

**Euler's formula** is an equation worked out by mathematician Leonhard
Euler regarding **complex numbers** that is instrumental to
the applied mathematics of physics, engineering, and related disciplines.
It relates exponentials to trigonometric functions, yielding a means
to convert an equation of trigonometric functions into one of
exponential functions, and vice versa, thus giving the mathematician
the choice of the methods of working with a function, such as
solving for some variable of interest or calculating with the
function. The equation:

*e*^{ix} = cos x + *i* sin x

*e* - **Euler's number**, a particular irrational number that begins 2.7182...
*i* - the square root of minus 1.
- sin, cos - trigonometric functions.
- x - any real number.

The formula was worked out using **series expansions**, a method of
comparing such functions and finding out how they are related, which
showed that given the notion of *i*, this formula produced
consistent and useful results. Among the formula's implications is
this equation:

e^{iπ} = -1
or
e^{iπ} + 1 = 0

This latter form of this equation is known as **Euler's identity**.

(*mathematics*)
**Further reading:**

http://en.wikipedia.org/wiki/Euler's_formula

**Referenced by pages:**

Fourier series

Fourier series expansion

Index