Gamma and the Pi function

The gamma function satisfies

Γ(n + 1) = n!

for all non-negative integers n, and extends to an analytic function in the complex plane with the exception of singularities at the non-positive integers [1]. Incidentally, going back to the previous post, this is an example of a theorem that would have to be amended if 0! were not defined to be 1.

Wouldn’t it be simpler if the gamma function were defined so that it’s value at n, not at n + 1, extended factorial? Well, some things would be simpler, but other things would not.

The Pi function defined by

Π(z) = Γ(z + 1)

extends factorial with no extra factor of 1 to keep up with, and some theorems are simpler to state in terms of the Pi function than the gamma function. For example, it’s simpler to say that the volume of a unit n-sphere is

(π/2)n/2 / Π(n/2)

than to say it’s

(π/2)n/2 / Γ(n/2 + 1).

The former has an easy-to-remember form, with lower case π on top and capital π on bottom.

The reflection identity is also a little simpler in the form

Π(z) Π(−z) = 1/sinc(z)

than in the form

Γ(z) Γ(1 − z) = π / sin(πz)

The drawback to the former is that you have to remember to use the convention

sinc(z) = sin(πz) / πz

because some sources define sinc with the factor of π and some without. Neither convention makes a large majority of theorems simpler, so there’s no clear winner. [2]

Fortunately the Pi function has a different name and isn’t an alternative convention for defining the gamma function. That would be terribly confusing. [3]

Related posts

[1] The Pi function is the unique way to extend factorial to an analytic function, given some regularity assumptions. See Wielandt’s theorem and the Bohr-Mollerup theorems.

[2] Fourier transforms may be the worst example in mathematics of no clear convention winner. Multiple conventions thrive because each makes some things easier to state. (See this page for a sort of Rosetta Stone translating between the various Fourier transform conventions.) In fact, the lack of standardization for the sinc function is related to the lack of standardization for the Fourier transform: you’d like to define the sinc function so that it has the simplest Fourier transform under your convention.

[3] I’m fuzzy on the history of all this, but I think what we now call the Pi function was an alternative definition of the gamma function briefly, but fortunately that got ironed out quickly.