Volume of an n-ball

In geometry, a ball is a region in space comprising all points within a fixed distance from a given point; that is, it is the region enclosed by a sphere or hypersphere. An n-ball is a ball in n-dimensional Euclidean space. The volume of an n-ball is an important constant that occurs in formulas throughout mathematics; it generalizes the notion of the volume enclosed by a sphere in 3-dimensional space. In geometry, a ball is a region in space comprising all points within a fixed distance from a given point; that is, it is the region enclosed by a sphere or hypersphere. An n-ball is a ball in n-dimensional Euclidean space. The volume of an n-ball is an important constant that occurs in formulas throughout mathematics; it generalizes the notion of the volume enclosed by a sphere in 3-dimensional space. The n-dimensional volume of a Euclidean ball of radius R in n-dimensional Euclidean space is: where Γ is Leonhard Euler's gamma function. The gamma function 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 · π1/2 if n is a non-negative integer. 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 that do not require an evaluation of the gamma function. These are: In the formula for odd-dimensional volumes, the double factorial (2k + 1)!! is defined for odd integers 2k + 1 as (2k + 1)!! = 1 · 3 · 5 · … · (2k − 1) · (2k + 1). Instead of expressing the volume V of the ball in terms of its radius R, the formula can be inverted to express the radius as a function of the volume: