Gamma Function: Properties & Examples

Instructor: Damien Howard

Damien has a master's degree in physics and has taught physics lab to college students.

Explore the properties of the gamma function including its ability to be represented in integral and factorial forms. Then dive deeper into the gamma function's properties by looking at several examples of them.

Improper Integrals

While you learn integration you'll work with two main types of integrals, definite integrals and indefinite integrals. The difference between the two is that definite integrals have limits of integration, and indefinite integrals do not. When you first start working with definite integrals you'll use simple real numbers for your limits of integration, but you'll soon see that more is possible. It's very common to see upper limits of integration set to infinity, and lower limits set to negative infinity.

Examples of an indefinite integral, a proper definite integral, and an improper definite integral
integral types

Any definite integral that has one or more infinite limits of integration, or an integrand that approaches infinity within its limits of integration is known as an improper integral. Improper integrals pop up all the time in calculus and other higher level math courses. In this lesson we'll look at the properties of one famous function defined by an improper integral known as the gamma function.

Euler's Integrals

The gamma function is defined for x > 0 in integral form by the improper integral known as Euler's integral of the second kind.

gamma function integral

As the name implies, there is also a Euler's integral of the first kind. This integral defines what is known as the beta function. However, the beta function can also be viewed as a combination of gamma functions.

beta function

An example of where you might see both the gamma and beta functions is within the field of statistics. These come up within the gamma and beta distributions that you'll work with often there.

Gamma Factorial Connection

While it's standard to define the gamma function in integral form by Euler's integral of the second kind, it can also be viewed as an extension of the factorial function when n is a positive integer.

gamma function factorial

This method is much nicer to work with than the Euler integral when possible, because it's generally quicker to solve a simple factorial than an integral. Let's look at a quick example where n = 6.

gamma function example

In order to show that this is indeed an accurate way to represent the gamma function we'll work through a proof of it together. Before we get started with the proof, there is one preliminary piece of knowledge that we need. It is that the following is true for n > 1.

gamma function preliminary

Now, to start working through the proof of the property we expand out Γ(n) as follows.

gamma proof part1

We now have Γ(1) multiplied by a large coefficient. This coefficient, as it turns out, is the definition for (n-1)!.

gamma proof part2

Therefore, the following is true.

gamma proof part3

Next, we need to show that Γ(1) = 1, and this can be done by working through Euler's integral of the second kind.

gamma proof part4

Finally, we complete the proof by substituting in the results for Γ(1) into the Γ(n) formula.

gamma proof part5

Other Common Properties

Along with the factorial function, there are few other prominent properties of the gamma function worth viewing. The first of which is as follows.

gamma function property1

We can see an example of this property working through a simple comparison of it to the standard factorial form of the gamma function with n = 4.

property1 example

So far we've primarily been working with the gamma function using integers since we've focused on its factorial form. The next three properties will show the gamma function operating on fractions and imaginary numbers, starting with the property that tells us what we get when solving for Γ(1/2).

gamma function property2

This property is actually a special case of the third property we'll go over, which finds the gamma function of any integer over two.

gamma function property3

You can see that when n = 1, we get the square root of π.

second and third property connection

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