The Gamma function, , is a curious mathematical creature that pops up in many, seemingly unrelated places. Yet it defies naive intuition;

  1. Over the positive integers, it simplifies to the factorial function, shifted by .
  2. Over the negative integers, it diverges.
  3. At , it diverges.
  4. It has even more interesting behavior over non-integer fractions and complex numbers.

What makes so interesting?

The Gamma function is defined as follows:

It converges for all complex numbers with positive real part (i.e. ). It also converges for complex numbers with negative real part (i.e. ) except where is a negative integer or .

When is zero or a negative integer and the imaginary part of is , then diverges and is undefined.

Note that the integers are embedded in the complex numbers as the subset obtained by setting the imaginary part to and restricting the real part accordingly.

Over the positive integers, has two general properties, outlined below.

Special Case

Consider for . The integral has a surprisingly simple simplification:

General Case: Positive Integers

Take to be an arbitrary positive integer, i.e. and . To prove that is indeed equivalent to the factorial function (albeit shifted by ), we need to show that .

Revisit the definition of , applied to :

This integral can be simplified using the rule for integration by parts:

Take and ; then:

This is an important identity, usually expressed in any of the following two forms:

Proof by Induction

So far, we have seen the following results:

By induction on , suppose we know that (with the base-case being ). We can compute as follows:


was not intended to be a generalization of the factorial function. The result was a somewhat accidental one, albeit useful.

By restricting to the positive integers, we discovered that:

But what happens when we plug in a fraction, say ?

To simplify the integral, let's try to get rid of the term by substituting :

This is a familiar integral, known as the Gaussian integral:

Since restricts the integration to and the function is even:

Other fractions of the form have similar results simplifications:

In general,

where is the Euler-Mascheroni constant and denotes asymptotic equivalence.

but unfortunately the integral has to be computed or simplified directly for each case.

has a meromorphic extension to the complex numbers, with simple poles at the non-positive integers and . It is defined with the same rules and the relation

holds for all complex numbers . For example;

Various values of are tabulated here.

For some useful applications of in the Riemann zeta function, see this post.