Euler's formula

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function.

where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively.

The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics".

In 1714, the English mathematician Roger Cotes presented a geometrical argument that can be interpreted (after correcting a misplaced factor of

[6][4] The formula was first published in 1748 in his foundational work Introductio in analysin infinitorum.

From a more advanced perspective, each of these definitions may be interpreted as giving the unique analytic continuation of ex to the complex plane.

Using the ratio test, it is possible to show that this power series has an infinite radius of convergence and so defines ez for all complex z.

Here, n is restricted to positive integers, so there is no question about what the power with exponent n means.

Here is a proof of Euler's formula using power-series expansions, as well as basic facts about the powers of i:[11]

where in the last step we recognize the two terms are the Maclaurin series for cos x and sin x.

Another proof[12] is based on the fact that all complex numbers can be expressed in polar coordinates.

From any of the definitions of the exponential function it can be shown that the derivative of eix is ieix.

Here φ is the angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counterclockwise and in radians.

In fact, the same proof shows that Euler's formula is even valid for all complex numbers x.

The polar form simplifies the mathematics when used in multiplication or powers of complex numbers.

where φ is the argument of z, i.e., the angle between the x axis and the vector z measured counterclockwise in radians, which is defined up to addition of 2π.

This is because for any real x and y, not both zero, the angles of the vectors (x, y) and (−x, −y) differ by π radians, but have the identical value of tan φ = ⁠y/x⁠.

These formulas can even serve as the definition of the trigonometric functions for complex arguments x.

Complex exponentials can simplify trigonometry, because they are mathematically easier to manipulate than their sine and cosine components.

One technique is simply to convert sines and cosines into equivalent expressions in terms of exponentials sometimes called complex sinusoids.

This formula is used for recursive generation of cos nx for integer values of n and arbitrary x (in radians).

Considering cos x a parameter in equation above yields recursive formula for Chebyshev polynomials of the first kind.

In the language of topology, Euler's formula states that the imaginary exponential function

These observations may be combined and summarized in the commutative diagram below: In differential equations, the function eix is often used to simplify solutions, even if the final answer is a real function involving sine and cosine.

In electrical engineering, signal processing, and similar fields, signals that vary periodically over time are often described as a combination of sinusoidal functions (see Fourier analysis), and these are more conveniently expressed as the sum of exponential functions with imaginary exponents, using Euler's formula.

Also, phasor analysis of circuits can include Euler's formula to represent the impedance of a capacitor or an inductor.

For any point r on this sphere, and x a real number, Euler's formula applies:

The special cases that evaluate to units illustrate rotation around the complex unit circle: The special case at x = τ (where τ = 2π, one turn) yields eiτ = 1 + 0.

This is also argued to link five fundamental constants with three basic arithmetic operations, but, unlike Euler's identity, without rearranging the addends from the general case:

An interpretation of the simplified form eiτ = 1 is that rotating by a full turn is an identity function.