III.31 The Gamma Function

Ben Green

If n is a positive integer, then its factorial, written n!, is the number 1 × 2 × · · · × n: that is, the product of all positive integers up to n. For example, the first eight factorials are 1, 2, 6, 24, 120, 720, 5040, and 40320. (The exclamation mark was introduced by Christian Kramp 200 years ago as a convenience to the printer: it is perhaps also intended to convey some alarm at the rapidity with which n! grows. An obsolete notation, which can still be found in some twentieth-century texts, is .) From this definition, it might appear to be impossible to make sense of the idea of the factorial of a number that is not a positive integer, but, as it turns out, it is not just possible to do so, but also extremely useful.

The gamma function, written Γ, is a function that agrees with the factorial function at positive integer values, but that makes sense for any real number, and even for any complex number. Actually, for various reasons it is natural to define Γ so that Γ(n) = (n- 1)! for n = 2, 3, . . . . Let us start by writing

without paying too much attention to whether the integral converges. If we integrate by parts, then we find that

As x tends to infinity, x^s-1e^-x tends to zero, and if s is, for example, a real number greater than 1, then x^s-1 = 0 when x = 0. Therefore, for such s, we can ignore the first term in the above expression. But the second one is simply the formula for Γ(s - 1), so we have shown that Γ(s) = (s - l)Γ(s - 1), which is just what we need if we want to think of Γ(s) as something like (s - 1)!.

It is not hard to show that the integral is in fact convergent whenever s is a complex number and Re(s) (the real part of s) is positive. Moreover, it defines a HOLOMORPHIC FUNCTION [I.3 §5.6] in that region. When the real part of s is negative, the integral does not converge at all, and so the formula (1) cannot be used to define the gamma function in its entirety. However, we can instead use the property Γ(s) = (s- l)Γ(s- 1) to extend the definition. For example, when -1 < Re(s) ≤ 0, we know that the definition does not work directly, but it does work for s + 1, since Re(s + 1) > 0. We would like Γ(s + 1) to equal sΓ(s), so it makes sense to define Γ (s) to be Γ(s + 1)/s. Once we have done this, we can turn our attention to values of s with -2 < Re(s) ≤ -1, and so on.

The reader may object that in defining Γ(0) (for example), we have divided by zero. This is perfectly permissible, however, if all we require of Γ is that it should be MEROMORPHIC [V.31], because meromorphic functions are allowed to take the “value” ∞. Indeed, it is not hard to see that Γ, as we have defined it, has simple poles at 0, -1, -2, . . . .

There are in fact many functions that share the useful properties of Γ. (For instance, because cos(2πs) = cos(2π(s + 1)) for any s, and cos(2πn) = 1 for every integer n, the function F(s) = Γ(s) cos(2πs) also has the property F(s) = (s-1)F(s-1)and F(n) = (n-1)!.) Nevertheless, for a variety of reasons, the function Γ, as we have defined it, is the most natural meromorphic extension of the factorial function. The most persuasive reason is the fact that it arises so often in natural contexts, but it is also, in a certain sense, the smoothest interpolation of the factorial function to all positive real values. In fact, if f : (0,∞) →, (0, ∞) is such that f (x + 1) = xf (x), f (1) = 1, and log f is convex, then f = Γ.

There are many interesting formulas involving Γ, such as Γ(s)Γ (1 - s) = π/sin(πs). There is also the famous result Γ() = , which is essentially equivalent to the fact that the area under the “normal distribution curve” h(x) = is 1 (this can be seen by making the substitution x = u²/2 in (1)). A very important result concerning Γ is the Weierstrass product expansion, which states that

for all complex z, where γ is Euler’s constant:

This formula makes it clear that Γ never vanishes, and that it has simple poles at 0 and the negative integers.

Why is the gamma function important? A simple reason is that it occurs frequently in many parts of mathematics, but one can still ask why this should be so. One reason is that Γ, as defined in (1), is the Mellin transform of the unarguably natural function f (x) = e^-x. The Mellin transform is a type of FOURIER TRANSFORM [III.27], but it is defined for functions on the group (⁺, ×) rather than (, +) (which is the habitat of the most familiar type of Fourier transform). For this reason, Γ is often seen in number theory, particularly ANALYTIC NUMBER THEORY [IV.2], where multiplicatively defined functions are often studied by taking Fourier transforms.

One appearance of Γ in a number-theoretical context is in the functional equation for the RIEMANN ZETA FUNCTION [IV.2 §3], namely,

Ξ(s) = Ξ(1 - s),

where

The ζ function has a well-known product representation

where the product is over primes and the representation is valid for Re (s) > 1. The extra factor Γ (s/2)π^-s/2 in (3) may be regarded as coming from the “prime at infinity” (a term which may be rigorously defined).

Stirling’s formula is a very useful tool in dealing with the gamma function: it provides a rather accurate estimate for Γ(z) in terms of simpler functions. A very rough (but often useful) approximation for n! is (n/e)ⁿ, which tells us that log(n!) is about n(log n - 1). Stirling’s formula is a sharper version of this crude estimate. Let δ > 0 and suppose that z is a complex number that has modulus at least 1 and argument between - π + δ and π - δ. (This second condition keeps z away from the negative real axis, where the poles are.) Then Stirling’s formula states that

where the error E is at most C (δ) / | z |. Here, C (δ) stands for a certain positive real number that depends on δ. (The smaller you make δ, the larger you have to make C (δ).) Using this, one may confirm that Γ decays exponentially as Im z → ∞ in any fixed vertical strip in the complex plane. In fact, if α < σ < β, then

for all | t | > 1, uniformly in σ.