23 - The Gamma Function

Summary

Preliminary Remarks

The problem of interpolating the sequence of factorials 0!, 1!, 2!, 3!, … appeared in Wallis's 1655 book on the quadrature of a circle. Specifically, the problem would be to find a function f(x) of a positive real variable x, such that f (x + 1) = x f (x) and f (n) = (n - 1)!, when n is a positive integer. One might wish to have f(n) = n! but we will identify f (x) with the gamma function Γ(x), so we shift the function f(n) = n! to the right by a unit. In his book, Wallis solved this problem for half-integral values of x but did not work out the details of the general case. Strangely, for seventy years after this, mathematicians did not take up this subject.

Euler and Stirling made significant contributions to this problem starting in the late 1720s. They worked independently, Euler in Russia and Stirling in Scotland; their approaches and aims were also distinct. In the mid-1730s, they came to know of each other's work and had a brief correspondence. Always the algorist, Euler was interested in obtaining analytic expressions for the interpolating function f(x). His first paper on the subject, written in 1730, gives two different representations of f(x), one as an infinite product and the other as a definite integral. On the other hand, Stirling was a numerical analyst interested in finding efficient methods for computing f(x).