Two results in mathematics are known as **Euler's formula**, after the mathematician Leonhard Euler.

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2 Complex analysis |

## Algebraic topology

In geometry and algebraic topology, there is a relationship called **Euler's formula** which relates the number of edges *E*, vertices *V*, and faces *F* of a simply connected polyhedron. Given such a polyhedron, the sum of the vertices and the faces is always the number of edges plus two. i.e.: *F* - *E* + *V* = 2.

### Derivation

For any polyhedron, it can be represented as a connected planar graph where *E*, *V* and *F* becomes the number of edges, vertices and disjointed area respectively. We can then prove that *F*-*E*+*V*=2 by mathematical induction on *E*.

### Generalisations

For arbitrary planar graphs, *F* - *E* + *V* - *C* = 1, where *C* is the number of components in the graph.

For nonplanar graphs that can be embedded in a manifold *M*, then *F* - *E* + *V* = χ(*M*), where χ is the Euler characteristic of the manifold, a constant which is invariant under continuous deformations. The Euler characteristic of a simply-connected manifold such as a sphere or a plane is 2. The proof of this generalisation requires the concept of triangulation.

## Complex analysis

In complex analysis, **Euler's formula**, attributed to the mathematician Leonhard Euler, states that

*x*. Here,

*e*is the base of the natural logarithm,

*i*is the imaginary unit and sin and cos are trigonometric functions.

This formula can be interpreted as saying that the function *e*^{ix} traces out the unit circle in the complex number plane as ranges through the real numbers. Here, is the angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counter clockwise and in radians. The formula is valid only if sin and cos take their arguments in radians rather than in degrees.

The proof is based on the Taylor series expansions of the exponential function *e*^{z} (where *z* is a complex number) and of sin *x* and cos *x* for real numbers *x* (see below). 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 (see Caspar Wessel).

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

In differential equations, the function *e*^{ix} is often used to simplify derivations, even if the final answer is a real function involving sine and cosine. Euler's identity 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 the real part of exponential functions with imaginary exponents, using Euler's formula.

### Derivation

Here is a derivation of Euler's formula using Taylor series expansions
plus some basic facts about *i*:

The function *e*^{x} (assuming *x* is a real) can be written as:

*x*is

*defined*by this series. Now if we throw

*i*into the exponent, we get:

*i*:

*n*:

*x*) and sin(

*x*):

*e*

^{ix}, gives: