There are several things in math and statistics named gamma. Three examples are

- the gamma
**function** - the gamma
**constant** - the gamma
**distribution**

This post will show how these are related. We’ll also look at the **incomplete** gamma function which connects with all the above.

## The gamma function

The **gamma function** is the most important function not usually found on a calculator. It’s the first “advanced” function you’re likely to learn about. You might see it in passing in a calculus class, in a homework problem on integration by parts, but usually not there’s not much emphasis on it. But it comes up a **lot **in application.

You can think of the gamma function as a way to extend factorial to non-integer values. For non-negative integers *n*, Γ(*n* + 1) = *n*!.

(Why is the argument *n* + 1 to Γ rather than *n*? There are a number of reasons, historical and practical. Short answer: some formulas turn out simpler if we define Γ that way.)

## The gamma constant

The **gamma constant**, a.k.a. **Euler’s constant** or the **Euler-Mascheroni constant**, is defined as the asymptotic difference between harmonic numbers and logarithms. That is,

The constant γ comes up fairly often in applications. But what does it have to do with the gamma *function*? There’s a reason the constant and the function are both named by the same Greek letter. One is that the gamma *constant* is part of the product formula for the gamma *function*.

If we take the logarithm of this formula and differentiation we find out that

## The gamma distribution

If you take the integrand defining the gamma *function* and turn it into a probability distribution by normalizing it to integrate to 1, you get the **gamma distribution**. That is, a gamma random variable with shape *k* has probability density function (PDF) given by

More generally you could add a scaling parameter to the gamma distribution in the usual way. You could imaging the scaling parameter present here but set to 1 to make things simpler.

## The incomplete gamma function

The **incomplete gamma function** relates to everything above. It’s like the (complete) gamma function, except the range of integration is finite. So it’s now a function of two variables, the extra variable being the limit of integration.

(Note that now *x* appears in the limit of integration, not the exponent of *t*. This notation is inconsistent with the definition of the (complete) gamma function but it’s conventional.)

It uses a lower case gamma for its notation, like the gamma *constant*, and is a generalization of the gamma *function*. It’s also essentially the cumulative distribution function of the gamma *distribution*. That is, the CDF of a gamma random variable with shape *s* is γ(*s*, *x*) / Γ(s).

The function γ(*s*, *x*) / Γ(s) is called the **regularized** incomplete gamma function. Sometimes the distinction between the regularized and unregularized versions is not explicit. For example, in Python, the function `gammainc`

does not compute the incomplete gamma function per se but the *regularized* incomplete gamma function. This makes sense because the latter is often more convenient to work with numerically.