Euler’s Formula is one of the most important formula’s in mathematics since it allows the simplification of many different ideas.

## Background

Named after Leonhard Euler:

- Born 15 April 1707
- Basel, Switzerland

- Death 18 September 1783 (aged 76)
- St. Petersburg, Russia

- Swiss mathematician and physicist
- Many important discoveries from infintesimal calculus to graph theory
- Especially in mathematical analysis - Euler’s formula
- Furthermore in many different physics and astronomy related topics

## The Idea

For any real number x:

It established a connection between the complex exponential function and the trigonometric funtions.

## The Proof

The Maclaurin Series (A special case of the Taylor Series) of the exponential function is:

With the basic facts about different powers of i:

And the Taylor Series of `\(\sin\)`

and `\(\cos\)`

:

We can proove:

## Applications

One of the best examples of this connection being used is the Fourier Series

## Sources

- http://www.gap-system.org/~history/Biographies/Euler.html
- http://books.google.com/books?id=PjK0F0T3NBoC&pg=PA428