Numb3rs: Euler's Formula

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

The Numb3rs Series

Numb3rs used to be its own subsection on this site, where I wrote math & science tutorials. Unfortunately this never came to frution so I integrated ths articles with the rest of the blog. If you want to check out the other Numb3rs posts, you can get an overview here.

By Cecil Wöbker


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