# Euler's identity

**Euler's Formula** is . It is named after the 18th-century mathematician Leonhard Euler.

## Contents

## Background

Euler's formula is a fundamental tool used when solving problems involving complex numbers and/or trigonometry. Euler's formula replaces "cis", and is a superior notation, as it encapsulates several nice properties:

### De Moivre's Theorem

De Moivre's Theorem states that for any real number and integer , .

### Sine/Cosine Angle Addition Formulas

Start with , and apply Euler's forumla both sides:

Expanding the right side gives

Comparing the real and imaginary terms of these expressions gives the sine and cosine angle-addition formulas:

### Geometry on the complex plane

### Other nice properties

A special, and quite fascinating, consequence of Euler's formula is the identity , which relates five of the most fundamental numbers in all of mathematics: e, i, pi, 0, and 1.

## Proof 1

The proof of Euler's formula can be shown using the technique from calculus known as Taylor series.

We have the following Taylor series:

The key step now is to let and plug it into the series for . The result is Euler's formula above.

## Proof 2

Define . Then ,

; we know , so we get , therefore .