## Monday Math 22: The Gamma Function (Part 5/5)

Part 5: n-ball
(Part 1)
(Part 2)
(Part 3)
(Part 4)

In mathematics, a ball is the region of three dimensional space bounded by a sphere. The concept of the sphere can be generalized to an n-sphere, which is the equivalent in n-dimensional space by the geometer’s convention, and the equivalent in (n+1)-dimensional space by the topologist’s convention; for clarity, we shall use the former convention. And so we can similarly define the n-ball: the closed n-ball with center  and radius R is the set of points in n-dimensional space (with our coordinates for the space named ) which obey the inequality . (If the inequality is instead strictly less than, we have the open n-ball).
Thus, the 1-ball is the line segment from  to ; the 2-ball is a disk; the 3-ball is the standard ball; the 4-ball is the interior of a hypersphere; and so on.
Now, let us consider the “volume”  of the n-ball, the n-dimensional measure of the region. We see from the above examples that , , and . In fact, geometry tells us that  must be proportional to ; thus, we define  such that .

Now, consider if we take an n-ball and expand the radius R by a tiny amount dR. Then we have increased it’s measure from  to . However, we can also see this as adding a thin “n-spherical” shell to the n-ball. When dR is infinitesimal, we see the volume dVn of the shell is the thickness dR times the n-1 dimensional measure  of the n-sphere of radius R (the “surface area” of the n-ball).
Thus , and so , and so we see that since , .

Next, let us consider an n-dimensional Gaussian, given by . Now, if we integrate this over all of n-dimensional space using our Cartesian coordinates, we see:
.

But this function also has n-spherical symmetry; it is a function of the distance  from the origin only. Thus, we see that this integral is equivalent to , which, using our formula , tells us that our integral is equal to
.
Let . Then , and making this substitution,
. Note that the integral in this last form is simply , and so:
.
Thus, , and .

Plugging in values for n confirms our formula is valid for n=1 through n=3; below is a table of the first few values of :

First few values of Cx)
n Cn
1 2
2 
3 
4 
5 
6 
7 
8 
9 
10 

Note that we always have an integer power of π. A quick examination of the formula and of the table, and one will see that when n is even (and thus  is an integer), we have . Using the formula  we found for the gamma function of half-integer values in part 1, we see that for odd n, .