In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the
spherical mean of a
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
around a point is the average of all values of that function on a sphere of given radius centered at that point.
Definition
Consider an
open setThe concept of an open set is fundamental to many areas of mathematics, especially pointset topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
U in the
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and threedimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
R^{n} and a
continuous functionIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
u defined on
U with
realIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
or
complexA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the onedimensional number line to the twodimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
values. Let
x be a point in
U and
r > 0 be such that the
closedIn geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
ballIn mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
B(
x,
r) of center
x and radius
r is contained in
U. The
spherical mean over the sphere of radius
r centered at
x is defined as

where ∂
B(
x,
r) is the (
n−1)sphere forming the
boundaryIn topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
of
B(
x,
r), d
S denotes integration with respect to
spherical measureIn mathematics — specifically, in geometric measure theory — spherical measure σn is the “natural” Borel measure on the nsphere Sn. Spherical measure is often normalized so that it is a probability measure on the sphere, i.e...
and
ω_{n−1}(
r) is the "surface area" of this (
n−1)sphere.
Equivalently, the spherical mean is given by

where
ω_{n−1} is the area of the (
n−1)sphere of radius 1.
The spherical mean is often denoted as

Properties and uses
 From the continuity of it follows that the function

 is continuous, and its limit
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....
as is
 Spherical means are used in finding the solution of the wave equation
The wave equation is an important secondorder linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
for with prescribed boundary conditions at

 This result can be used to prove the maximum principle
In mathematics, the maximum principle is a property of solutions to certain partial differential equations, of the elliptic and parabolic types. Roughly speaking, it says that the maximum of a function in a domain is to be found on the boundary of that domain...
for harmonic functions.