The Euler formula states

(1) |

(2) |

(3) |

The Euler formula can be demonstrated using a series expansion

(4) |

It can also be proven using a Complex integral. Let

(5) |

(6) |

(7) |

(8) |

(9) |

**References**

Castellanos, D. ``The Ubiquitous Pi. Part I.'' *Math. Mag.* **61**, 67-98, 1988.

Conway, J. H. and Guy, R. K. ``Euler's Wonderful Relation.'' *The Book of Numbers.* New York: Springer-Verlag,
pp. 254-256, 1996.

Cotes, R. *Philosophical Transactions* **29**, 32, 1714.

Euler, L. *Miscellanea Berolinensia* **7**, 179, 1743.

Euler, L. *Introductio in Analysin Infinitorum, Vol. 1.* Lausanne, p. 104, 1748.

© 1996-9

1999-05-25