hypersphere_integrals, an Octave code which returns the exact value of the integral of any monomial over the surface of the unit hypersphere in M dimensions.
The surface of the unit hypersphere in M dimensions is defined by
sum ( 1 <= i <= m ) x(i)^2 = 1
The integrands are all of the form
f(x) = product ( 1 <= m <= m ) x(i) ^ e(i)where the exponents e are nonnegative integers. If any exponent is an odd integer, the integral will be zero. Thus, the "interesting" results occur when all exponents are even.
The computer code and data files described and made available on this web page are distributed under the MIT license
hypersphere_integrals is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.
ball_integrals, an Octave code which returns the exact value of the integral of any monomial over the interior of the unit ball in 3D.
circle_integrals, an Octave code which returns the exact value of the integral of any monomial over the surface of the unit circle in 2D.
cube_integrals, an Octave code which returns the exact value of the integral of any monomial over the interior of the unit cube in 3D.
disk01_integrals, an Octave code which returns the exact value of the integral of any monomial over the interior of the unit disk in 2D.
hexagon_integrals, an Octave code which returns the exact value of the integral of a monomial over the interior of a hexagon in 2D.
hyperball_integrals, an Octave code which returns the exact value of the integral of any monomial over the interior of the unit hyperball in M dimensions.
hypercube_integrals, an Octave code which returns the exact value of the integral of any monomial over the interior of the unit hypercube in M dimensions.
line_integrals, an Octave code which returns the exact value of the integral of any monomial over the length of the unit line in 1D.
simplex_integrals, an Octave code which returns the exact value of the integral of any monomial over the interior of the unit simplex in M dimensions.