Factorials and the Gamma function

By Murray Bourne, 14 Apr 2010

In math, we often come across the following expression:

n!

This is "n factorial", or the product

n(n − 1)(n − 2)(n − 3) ... (3)(2)(1).

Factorials are used in the study of counting and probability. For example, permutations (which involves counting the arrangement of objects where the order is important) and combinations (where the order is not important) both require factorials when the number of objects is large.

cards 

Also, finding the probability of winning Lotto or cards also involves factorials.

Examples of factorials:

2! = 2 × 1 = 2

3! = 3 × 2 × 1 = 6

4! = 4 × 3 × 2 × 1 = 24

5! = 5 × 4 × 3 × 2 × 1 = 120




Product Notation

We can write factorials using product notation (upper case "pi") as follows:

n!=\prod_{k=1}^n{k}\!

This notation works in a similar way to summation notation (Σ), but in this case we multiply rather than add terms. For example, if n = 4, we would substitute k = 1, then k = 2, then k = 3 and finally k = 4 and write:

4!=\prod_{k=1}^4{k}\!=1\times2\times3\times4=24 

What about 0! ?

0! is a special case. There are many situations where the value of 0! has a meaning, and it has value 1 in those cases, not 0. So by convention, we define:

0! = 1

Also, 1! has value 1:

1! = 1

The graph of f(n) = n! for integer values n is as follows:

graph of factorial values

Note that points A(0,1), B(1,1), C(2,2), D(3,6), E(4,24), and F(5, 120) are discrete points in space - we have not connected them with a curve. There is no meaning for non-integer factorials like the expression 3.5! (3.5 factorial).

However, our graph does suggest a curve - one that is (approximately) exponentially increasing.

Is there a function we can use that fits this curve and so gives us meaningful values for factorials of numbers which are not whole numbers?

It turns out there is.

The Gamma Function

The Gamma Function is an extension of the concept of factorial numbers. We can input (almost) any real or complex number into the Gamma function and find its value. Such values will be related to factorial values.

There is a special case where we can see the connection to factorial numbers.

If n is a positive integer, then the function Gamma (named after the Greek letter "Γ" by the mathematician Legendre) of n is:

Γ(n) = (n − 1)!

We can easily "shift" this by 1 and obtain an expression for n! as follows:

Γ(n + 1) = n!

But the Gamma function is not restricted to the whole numbers (that's the point). A formula that allows us to find the value of the Gamma function for any real value of n is as follows:

\Gamma(n+1)=\int_0^{\infty}{e^{-x}}{x^n}dx

For example, let n = 3.5. We want to find the value of 3.5!, assuming it exists.

The value of Γ(3.5 + 1) = Γ(4.5) is given by the infinite integral:

\Gamma(4.5)=\int_0^{\infty}{e^{-x}}{x^{3.5}}dx 

Examining the graph above, we expect this value to be somewhere in the range 10 to 15. (Recall that 3! = 6 and 4! = 24. Our answer for Γ(4.5) = 3.5! has to be between these values.)

What does this integral mean?

The function under the integral sign is very interesting. It is the product of an ever-decreasing function with an ever-increasing one.

f(x) = exx3.5

Let's look at the graphs involved in this expression.

Firstly, f(x) = ex. Note that the value of the function (its height) decreases as x increases.

graph of e^-x

Secondly, f(x) = x3.5 increases as x increases. This function is not defined (over the reals) for negative x.

graph of x^3.5

Finally, we look at the product of the 2 functions. This is the graph of f(x) = exx3.5.

graph of e^-xx^3.5

The area under this graph (the shaded portion) from 0 to ∞ (infinity) gives us the value of Γ(4.5) = 3.5!.

We use computer mathematics software (Scientific Notebook) to find the value of the integral. We only need to choose some "large number" (I chose 10000) for the upper bound of the integral since as you can see in the graph, the height of the curve is very small as x becomes very large. Here's what we get:

\Gamma(4.5)=\int_0^{10000}{e^{-x}}{x^{3.5}}dx\approx11.6317

This means the shaded area above is 11.6317 square units. This is in the range we estimated earlier.

Back to the Graph

Let's return to our graph of the values of factorial numbers. We can use the above integral to calculate values of the Gamma function for any real value of n.

This time, I have included a smooth curve passing through our factorial values. This curve is f(n) = Γ(n + 1).

I've also added the new point F(3.5, 11.6317), which is the ordered pair representing Γ(4.5) = 11.6317 we just found. You can see this new point lies on the smooth curve joining the other factorial values.

Gamma 3.5

You can see more information on Γ(4.5) using Wolfram | Alpha.

Conclusion

The Gamma function gives us values that are analogous to factorials of non-integer numbers. It was one of the many brilliant contributions to the world of math by the Swiss mathematician Leonhard Euler.

To finish, let's look at the graph of f(n) = Γ(n + 1) for a greater range of real values of n. We observe there are some "holes" (discontinuities) in the graph for the negative integers.

Gamma function

See the 17 Comments below.

17 Comments on “Factorials and the Gamma function”

  1. anantha murthy says:

    This method of learning is simply fantastic.Please do help me keep going with this method of teaching.

  2. sesham srinu says:

    excellent explanation
    thank you for sending this doc

  3. m bose says:

    exquisitely simple...great reading for grasping the basic concept !

  4. ramesh says:

    hi realy i dont know how to find the factorial value of fractional numbers. now only i know the real uses of gamma funciton. thank very much.

  5. Sukhbir Singh says:

    really good one...
    gamma function from area point of view is really amazig...
    if some one can explain me the difference betweeen different transforms like fourier, laplace, Z ...it will be a great help...
    thanks in advance...

  6. Dibyendu De says:

    The way Gamma function was explained is simply fascinating.

  7. Luther McFarlane says:

    Brilliant explanation of finding the value of fractional factorials using gamma function. Thank you very much.

  8. claudio says:

    How Euler found this integral to resolve the problem. What was on his brain during this brilliant process.- We always miss the most important features and conforms only with the outputs

  9. kulaithem says:

    Very nice topic =)
    Thanks a lot ++

  10. arunpatil says:

    gamma function is funtastics way to caculate factorial of fractional number but some like -2.3 then how

  11. Murray says:

    @Aunpatil: You can see factorial values for negative numbers in the last graph in the article. Note there are some discontinuities.

  12. R.Kesavan says:

    Dear Sir,

    The most complex and difficult concepts of higher mathematics is explained and elucidated in a lucid, crystal clear way that could easily inculcate into lesser ignoramuses and laggards. What a splendid way of pedagogy that makes the gap between the pedagogue and the pupil becoming less and lesser.
    Kindly accept my kudos and encomiums for such a beautiful explanation making such gems of teachers a rare breed, nay oasis in the desert !!!.
    R.Kesavan.
    kesavan7777@yahoo.com

  13. Murray says:

    @R. Kesevan: Thank you!

  14. Lum says:

    Thanks for making learning easier!!!

  15. Sadiq Akbar says:

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  16. Daedalus says:

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  17. Abdellah says:

    Best explanation i found in internet.
    Thank you

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