# hypersphere_integrals

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.

### Languages:

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.

### Related Data and Programs:

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.

### Reference:

1. Gerald Folland,
How to Integrate a Polynomial Over a Sphere,
American Mathematical Monthly,
Volume 108, Number 5, May 2001, pages 446-448.

### Source Code:

Last revised on 01 February 2019.