# hypersphere_integrals

hypersphere_integrals, a C 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 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 a Python version.

### Related Data and Programs:

BALL_INTEGRALS, a C code which returns the exact value of the integral of any monomial over the interior of the unit ball in 3D.

CIRCLE_INTEGRALS, a C code which returns the exact value of the integral of any monomial over the surface of the unit circle in 2D.

CUBE_INTEGRALS, a C code which returns the exact value of the integral of any monomial over the interior of the unit cube in 3D.

DISK01_INTEGRALS, a C code which returns the exact value of the integral of any monomial over the interior of the unit disk in 2D.

HYPERBALL_INTEGRALS, a C code which returns the exact value of the integral of any monomial over the interior of the unit hyperball in M dimensions.

HYPERCUBE_INTEGRALS, a C code which returns the exact value of the integral of any monomial over the interior of the unit hypercube in M dimensions.

HYPERSPHERE_MONTE_CARLO, a C code which uses the Monte Carlo method to estimate the integral of a function over the surface of the unit hypersphere in M dimensions.

HYPERSPHERE_PROPERTIES, a C code which carries out various operations for an M-dimensional hypersphere, including converting between Cartesian and spherical coordinates, stereographic projection, sampling the surface of the sphere, and computing the surface area and volume.

LINE_INTEGRALS, a C code which returns the exact value of the integral of any monomial over the length of the unit line in 1D.

POLYGON_INTEGRALS, a C code which returns the exact value of the integral of any monomial over the interior of a polygon in 2D.

PYRAMID_INTEGRALS, a C code which returns the exact value of the integral of any monomial over the interior of the unit pyramid in 3D.

SIMPLEX_INTEGRALS, a C code which returns the exact value of the integral of any monomial over the interior of the unit simplex in M dimensions.

SPHERE_INTEGRALS, a C code which returns the exact value of the integral of any monomial over the surface of the unit sphere in 3D.

SQUARE_INTEGRALS, a C code which returns the exact value of the integral of any monomial over the interior of the unit square in 2D.

TETRAHEDRON_INTEGRALS, a C code which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TRIANGLE_INTEGRALS, a C code which returns the exact value of the integral of any monomial over the interior of a triangle in 2D.

WEDGE_INTEGRALS, a C code which returns the exact value of the integral of any monomial over the interior of the unit wedge in 3D.

### 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 09 July 2019.