hypersphere_integrals


hypersphere_integrals, a Python 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.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

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:

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triangle_integrals, a python code which returns the exact value of the integral of any monomial over the interior of the unit triangle in 2d.

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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 26 January 2020.