Ball

The -ball, denoted , is the interior of a sphere , and sometimes also called the -disk. (Although physicists often use the term "sphere" to mean the solid ball, mathematicians definitely do not!)

The ball of radius centered at point is implemented in the Wolfram Language as Ball[x, y, z, r].

The equation for the surface area of the -dimensional unit hypersphere gives the recurrence relation

(1)

Using then gives the hypercontent of the -ball of radius as

(2)

(Sommerville 1958, p. 136; Apostol 1974, p. 430; Conway and Sloane 1993). Strangely enough, the content reaches a maximum and then decreases towards 0 as increases. The point of maximal content of a unit -ball satisfies

			(3)
			(4)
			(5)

where is the digamma function, is the gamma function, is the Euler-Mascheroni constant, and is a harmonic number. This equation cannot be solved analytically for , but the numerical solution to

(6)

is (OEIS A074455) (Wells 1986, p. 67). As a result, the five-dimensional unit ball has maximal content (Le Lionnais 1983; Wells 1986, p. 60).

The following table gives the content for the unit radius -ball (OEIS A072345 and A072346), ratio of the volume of the -ball to that of a circumscribed hypercube (OEIS A087299), and surface area of the -ball (OEIS A072478 and A072479).

0	1	1	0
1	2	1	2
2
3
4
5
6
7
8
9
10

Let denote the volume of an -dimensional ball of radius . Then

			(7)
			(8)

so

(9)

where is the erf function (Freden 1993).

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