Gamma Function - Motivation

Motivation

The gamma function can be seen as a solution to the following interpolation problem:

"Find a smooth curve that connects the points (x, y) given by y = (x − 1)! at the positive integer values for x."

A plot of the first few factorials makes clear that such a curve can be drawn, but it would be preferable to have a formula that precisely describes the curve, in which the number of operations does not depend on the size of x. The simple formula for the factorial, n! = 1 × 2 × … × n, cannot be used directly for fractional values of x since it is only valid when x is a natural number (i.e., a positive integer). There are, relatively speaking, no such simple solutions for factorials; any combination of sums, products, powers, exponential functions, or logarithms with a fixed number of terms will not suffice to express x!. Stirling's approximation is asymptotically equal to the factorial function for large values of x. It is possible to find a general formula for factorials using tools such as integrals and limits from calculus. A good solution to this is the gamma function.

There are infinitely many continuous extensions of the factorial to non-integers: infinitely many curves can be drawn through any set of isolated points. The gamma function is the most useful solution in practice, being analytic (except at the non-positive integers), and it can be characterized in several ways. However, it is not the only analytic function which extends the factorial, as adding to it any analytic function which is zero on the positive integers will give another function with that property.

for x equal to any positive real number. The Bohr–Mollerup theorem proves that these properties, together with the assumption that f be logarithmically convex (aka: "superconvex"), uniquely determine f for positive, real inputs. From there, the gamma function can be extended to all real and complex values (except the negative integers and zero) by using the unique analytic continuation of f.