Volume of an n-ball

In geometry, a ball is a region in a space comprising all points within a fixed distance, called the radius, from a given point; that is, it is the region enclosed by a sphere or hypersphere. An n-ball is a ball in an n-dimensional Euclidean space. The volume of a n-ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume of a ball in 3-dimensional space. The volume of a n-ball of radius R is where is the volume of the unit n-ball, the n-ball of radius 1. The real number can be expressed via a two-dimension recurrence relation. Closed-form expressions involve the gamma, factorial, or double factorial function. The volume can also be expressed in terms of , the area of the unit n-sphere. The first volumes are as follows: The n-dimensional volume of a Euclidean ball of radius R in n-dimensional Euclidean space is: where Γ is Euler's gamma function. The gamma function is offset from but otherwise extends the factorial function to non-integer arguments. It satisfies Γ(n) = (n − 1)! if n is a positive integer and Γ(n + 1/2) = (n − 1/2) · (n − 3/2) · ... · 1/2 · pi1/2 if n is a non-negative integer. The volume can be computed without use of the Gamma function. As is proved below using a vector-calculus double integral in polar coordinates, the volume V of an n-ball of radius R can be expressed recursively in terms of the volume of an (n − 2)-ball, via the interleaved recurrence relation: This allows computation of V_n(R) in approximately n / 2 steps. The volume can also be expressed in terms of an (n − 1)-ball using the one-dimension recurrence relation: Inverting the above, the radius of an n-ball of volume V can be expressed recursively in terms of the radius of an (n − 2)- or (n − 1)-ball: Using explicit formulas for particular values of the gamma function at the integers and half-integers gives formulas for the volume of a Euclidean ball in terms of factorials. For non-negative integer k, these are: The volume can also be expressed in terms of double factorials.