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: \[e^{i x} = \cos{c} + i \sin{x}\]

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: \[e^{x} = \sum^{\infty}_{n=0} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\text{ for all } x\!\]

With the basic facts about different powers of i: \[\begin{align} i^0 &{}= 1, \quad & i^1 &{}= i, \quad & i^2 &{}= -1, \quad & i^3 &{}= -i, \\ i^4 &={} 1, \quad & i^5 &={} i, \quad & i^6 &{}= -1, \quad & i^7 &{}= -i, \end{align}\]

And the Taylor Series of \(\sin\) and \(\cos\): \[\sin x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\text{ for all } x\\] \[\cos x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\text{ for all } x\!\]

We can proove: \[\begin{align} e^{ix} &{}= 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \frac{(ix)^5}{5!} + \frac{(ix)^6}{6!} + \frac{(ix)^7}{7!} + \frac{(ix)^8}{8!} + \cdots \\ &{}= 1 + ix - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} + \frac{ix^5}{5!} - \frac{x^6}{6!} - \frac{ix^7}{7!} + \frac{x^8}{8!} + \cdots \\ &{}= \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \cdots \right) + i\left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \right) \\ &{}= \cos x + i\sin x \ . \end{align}\]

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