CIRCLE_INTEGRALS
Integrals Along the Circumference of the Unit Circle in 2D

CIRCLE_INTEGRALS is a C++ library which returns the exact value of the integral of any monomial along the circumference of the unit circle in 2D.

The circumference of the unit circle in 2D is defined by

        x^2 + y^2 = 1

The integrands are all of the form

        f(x,y) = x^e1 * y^e2

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.

Licensing:

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

Languages:

CIRCLE_INTEGRALS is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

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

CIRCLE_MONTE_CARLO, a C++ library which uses the Monte Carlo method to estimate the integral of a function over the circumference of the unit circle in 2D.

CIRCLE_RULE, a C++ library which computes quadrature rules for the unit circle in 2D, that is, the circumference of the circle of radius 1 and center (0,0).

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

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

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

HYPERSPHERE_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the surface of the unit hypersphere in M dimensions.

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

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

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

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

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

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

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

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

Reference:

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

Source Code:

circle_integrals.cpp, the source code.
circle_integrals.hpp, the include file.
circle_integrals.sh, BASH commands to compile the source code.

Examples and Tests:

circle_integrals_prb.cpp, a sample calling program.
circle_integrals_prb.sh, BASH commands to compile and run the sample program.
circle_integrals_prb_output.txt, the output file.

List of Routines:

CIRCLE01_LENGTH: length of the circumference of the unit circle in 2D.
CIRCLE01_MONOMIAL_INTEGRAL: integral on circumference of unit circle in 2D.
CIRCLE01_SAMPLE samples the circumference of the unit circle in 2D.
I4VEC_UNIFORM_AB returns a scaled pseudorandom I4VEC.
MONOMIAL_VALUE evaluates a monomial.
R8_GAMMA evaluates Gamma(X) for a real argument.
R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.

Last revised on 12 January 2014.

CIRCLE_INTEGRALS Integrals Along the Circumference of the Unit Circle in 2D