Euler’s Gamma Function and Ball/Cube Peg/Hole Problem

n!

No, I’m not really excited about the letter n. f(n) = n! is the factorial function. Taking the factorial of a positive integer n can be defined as follows:

n(n-1)(n-2)…(3)(2)(1)

A graph of the gamma function. Image: public domain, via Wikimedia Commons.

A graph of the gamma function. Image: public domain, via Wikimedia Commons.

Factorial has its uses in many areas of mathematics. Counting, permutations, the “n choose m” algorithm – all utilize factorial to a certain extent. Like me, some of you may have wondered why factorial cannot be extended to more than just the positive integers. Leonhard Euler’s gamma function resolves this problem.

John Wallis, who lived primarily in the 17th century, took the first steps in defining factorial outside of the positive integers. He knew the following two integrals were true:

expression 1

expression 2

Using the second formula and picking n = 1/2 (to match the first integral) yields:

expression 3

Solving for 1/2 ! reveals the following:

expression 4

This means that, assuming factorial can be defined outside of positive integers, 1/2 ! should be equal to √π/2.

In 1730, in a letter to Christian Goldbach, Euler defined n factorial for positive real numbers.

expression 5

If we replace n by t (signifying that n doesn’t have to be a natural number), and do the substitution x = -ln(s), we get the following:

expression 6

expression 7

The bound 0 becomes ∞ and the bound 1 becomes 0. Completing the substitution, we get this integral:

expression 8

Flipping the bounds so the smaller is the on the bottom requires multiplying by negative one.

expression 9

Mathematicians have defined the gamma function to shift the input variable down by one, so the modern gamma function (for positive values of t) is the following:

expression 10

If t is a positive integer, it can be defined more simply as follows:

expression 11

One interesting application of the Gamma function is the question, “Does a round peg fit better in a square hole than a square peg in a round hole?” This question can be simplified to a matter of ratios. That is, is the ratio of the area of the circle to that of the circumscribed square or the ratio of the area of the square to that of the circumscribed circle bigger? This question was considered by Jeffrey Nunemacher in 1986 in his article The Largest Unit Ball in any Euclidean Space. It laid the framework for Joel Azose’s work on it in his work, On the Gamma Function and its Applications. Azose used the following method to solve the problem using the gamma function.

If we expand the problem to the nth dimension, then the volume of the unit “n-ball” (a 2-ball would be a circle, a 3-ball a sphere) is as follows:

expression 12

Because the side length of a circumscribed n-cube (a 2-cube would be a square, a 3-cube a cube) is equal to the diameter of the n-ball, the volume of our circumscribed n-cube is:

expression 13

Because the diagonal of an inscribed n-cube is equal to √n times its edge as well as being the diameter of the circle, a side of the inscribed n-cube is 2/√n. Therefore its volume is:

expression 14

If the ratio of the volume of the n-ball to the volume of the circumscribed n-cube is R1(n) and the ratio of the volume of the inscribed n-cube to the volume of the n-ball is R2(n), we get:

expression 15

expression 16

We can then take the ratio of R1 to R2, which is:

expression 17

The ratio simplifies to this:

expression 18

As n approaches infinity, the ratio approaches zero. This is true because 22n easily outgrows πn/2, and although it is not obvious, the gamma function easily outgrows nn/2 when it is squared. Therefore, the denominator grows much more quickly than the numerator for large n, so the limit as n approches infinity is zero. This means that for sufficiently high values of n, the n-cube fits the n-ball better than the n-ball fits the n-cube. As it turns out, this is true for n≥9.

The gamma function is certainly one of the most interesting single variable functions I have seen in mathematics. Its graph seems very unusual, and it visually appears to contain parts of other functions’ graphs. I haven’t researched many applications, but this is definitely one of the more interesting ones I found. Plus, it has answered my question: what if the factorial function could take non integer inputs?

References:

https://en.wikipedia.org/wiki/Gamma_function

https://www.math.washington.edu/~morrow/336_10/papers/joel.pdf

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