C# Class Accord.Math.Bessel

Bessel functions.

Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are the canonical solutions y(x) of Bessel's differential equation.

Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are therefore especially important for many problems of wave propagation and static potentials. In solving problems in cylindrical coordinate systems, one obtains Bessel functions of integer order (α = n); in spherical problems, one obtains half-integer orders (α = n+1/2). For example:

Electromagnetic waves in a cylindrical waveguide Heat conduction in a cylindrical object Modes of vibration of a thin circular (or annular) artificial membrane (such as a drum or other membranophone) Diffusion problems on a lattice Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle Solving for patterns of acoustical radiation Frequency-dependent friction in circular pipelines

Bessel functions also appear in other problems, such as signal processing (e.g., see FM synthesis, Kaiser window, or Bessel filter).

This class offers implementations of Bessel's first and second kind functions, with special cases for zero and for arbitrary n.

References: Cephes Math Library, http://www.netlib.org/cephes/ Wikipedia contributors, "Bessel function,". Wikipedia, The Free Encyclopedia. Available at: http://en.wikipedia.org/wiki/Bessel_function

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Public Methods

Method Description
I ( double x ) : double

Bessel function of the first kind, of order 1.
I ( int n, double x ) : double

Bessel function of the first kind, of order n.
I0 ( double x ) : double

Bessel function of the first kind, of order 0.
J ( double x ) : double

Bessel function of order 1.
J ( int n, double x ) : double

Bessel function of order n.
J0 ( double x ) : double

Bessel function of order 0.
Y ( double x ) : double

Bessel function of the second kind, of order 1.
Y ( int n, double x ) : double

Bessel function of the second kind, of order n.
Y0 ( double x ) : double

Bessel function of the second kind, of order 0.

Method Details

I() public static method

Bessel function of the first kind, of order 1.
public static I ( double x ) : double
x double
return double

I() public static method

Bessel function of the first kind, of order n.
public static I ( int n, double x ) : double
n int
x double
return double

I0() public static method

Bessel function of the first kind, of order 0.
public static I0 ( double x ) : double
x double
return double

J() public static method

Bessel function of order 1.
public static J ( double x ) : double
x double
return double

J() public static method

Bessel function of order n.
public static J ( int n, double x ) : double
n int
x double
return double

J0() public static method

Bessel function of order 0.
public static J0 ( double x ) : double
x double
return double

Y() public static method

Bessel function of the second kind, of order 1.
public static Y ( double x ) : double
x double
return double

Y() public static method

Bessel function of the second kind, of order n.
public static Y ( int n, double x ) : double
n int
x double
return double

Y0() public static method

Bessel function of the second kind, of order 0.
public static Y0 ( double x ) : double
x double
return double