Factorials and the Gamma function
By Murray Bourne, 14 Apr 2010
In math, we often come across the following expression:
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.
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
We can write factorials using product notation (upper case "pi") as follows:
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:
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:
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:
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:
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) = e−xx3.5
Let's look at the graphs involved in this expression.
Firstly, f(x) = e−x. Note that the value of the function (its height) decreases as x increases.
Secondly, f(x) = x3.5 increases as x increases. This function is not defined (over the reals) for negative x.
Finally, we look at the product of the 2 functions. This is the graph of f(x) = e−xx3.5.
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:
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.
You can see more information on Γ(4.5) using Wolfram | Alpha.
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.
See the 19 Comments below.
25 Apr 2010 at 12:15 am [Comment permalink]
This method of learning is simply fantastic.Please do help me keep going with this method of teaching.
25 Apr 2010 at 8:36 pm [Comment permalink]
thank you for sending this doc
3 May 2010 at 6:52 pm [Comment permalink]
exquisitely simple...great reading for grasping the basic concept !
11 May 2010 at 9:51 pm [Comment permalink]
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.
17 May 2010 at 1:31 pm [Comment permalink]
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...
5 Sep 2010 at 9:04 pm [Comment permalink]
The way Gamma function was explained is simply fascinating.
24 Sep 2010 at 5:09 am [Comment permalink]
Brilliant explanation of finding the value of fractional factorials using gamma function. Thank you very much.
14 Apr 2011 at 10:03 pm [Comment permalink]
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
5 Jul 2011 at 11:13 pm [Comment permalink]
Very nice topic =)
Thanks a lot ++
30 Jul 2011 at 8:08 pm [Comment permalink]
gamma function is funtastics way to caculate factorial of fractional number but some like -2.3 then how
31 Jul 2011 at 7:22 am [Comment permalink]
@Aunpatil: You can see factorial values for negative numbers in the last graph in the article. Note there are some discontinuities.
21 Nov 2011 at 6:20 pm [Comment permalink]
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Kindly accept my kudos and encomiums for such a beautiful explanation making such gems of teachers a rare breed, nay oasis in the desert !!!.
21 Nov 2011 at 7:00 pm [Comment permalink]
@R. Kesevan: Thank you!
29 Nov 2012 at 11:50 am [Comment permalink]
Thanks for making learning easier!!!
11 Feb 2013 at 6:57 pm [Comment permalink]
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8 Dec 2015 at 10:10 pm [Comment permalink]
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6 Apr 2016 at 9:31 pm [Comment permalink]
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15 Sep 2019 at 10:17 pm [Comment permalink]
Excellent for a refresher or first introduction. Thank you for your clear approach.
2 Nov 2020 at 12:37 am [Comment permalink]
i just dont know how to predict the exact value of the integral which includes the function