# Abstract

To solve the problem of extending the factorial to non-integer arguments, Leonhard Euler gave a representation of the factorial as an infinite product. In this knol we give a derivation of this product.

**INTRODUCTION**

In the 1720’s, the best mathematicians of the day were interested in defining the factorial function for values other than nonnegative integers. Euler was the first to accomplish this; he did it by expressing the factorial as an infinite product:

In doing so, he made it possible to evaluate expressions like , which using the product is seen to be approximately 7.188082733.

**PROOF**

__ is clearly true, so in the following we will assume .__

__Proof__:

__Proof__:

__Proof__:

Notice that any positive value of n can be substituted in the infinite product, not just nonnegative integers. This representation led immediately to the very important gamma function.