Bessel function of the first kind of integer order.

Contents

Interface Function Documentation Page Comments

Group Description

These function return solutions to the Bessel Function of the first kind. They are defined as solutions to the Bessel differential equation which are both nonsingular at the origin. They are are sometimes called cylinder functions or cylindrical harmonics.

References:

http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

Interface

#include <codecogs/maths/special/bessel/j/j.h>

using namespace Maths::Special::Bessel::J;

	double	J (double x, int v)[inline] Bessel function of the first kind of integer order.
	Real	cc_J (Real x, Integer v) This function is available as a Microsoft Excel add-in.
	Real	cc_J_di (Real x, Integer v) This function is available as a Microsoft Excel add-in.
	double	J (double x, double v) Bessel function of the first kind.
	Real	cc_J_dd (Real x, Real v) This function is available as a Microsoft Excel add-in.

Function Documentation

Returns Bessel function of the first kinds for any integer order (v)

The ratio of jn(x) to j0(x) is computed by backward recurrence. First the ratio jn/jn-1 is found by a continued fraction expansion. Then the recurrence relating successive orders is applied until j0 or j1 is reached.

If n = 0 or 1 the routine for j0 or j1 is called directly.

Accuracy:

                           Relative error:
 arithmetic   domain     # trials      peak         rms
    DEC       0, 30        5500       6.9e-17     9.3e-18
    IEEE      0, 30        5000       4.4e-16     7.9e-17
 

Example:

#include <stdio.h>
#include <codecogs/maths/special/bessel/j/j.h>
int main()
{
  printf("\n  x      v=0      v=1      v=2      v=3      v=4      v=5");
  for(double x=0; x<6; x++)
  {
    printf("\nx=%.1lf",x);
    for(int v=0;v<=5;v++)
      printf(" %8.6lf", Maths::Special::Bessel::J::J(x,v));
  }
  return 0;
}

Output:

x      v=0      v=1      v=2      v=3      v=4      v=5
x=0.0 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000
x=1.0 0.765198 0.440051 0.114903 0.019563 0.002477 0.000250
x=2.0 0.223891 0.576725 0.352834 0.128943 0.033996 0.007040
x=3.0 -0.260052 0.339059 0.486091 0.309063 0.132034 0.043028
x=4.0 -0.397150 -0.066043 0.364128 0.430171 0.281129 0.132087
x=5.0 -0.177597 -0.327579 0.046565 0.364831 0.391232 0.261141

See also Maths/Special/Bessel/J/J1 and Maths/Special/Bessel/J/J0

Warnings:

Not suitable for large n or x. Use J() with real v instead.

References:

Cephes Math Library Release 2.8: June, 2000

Parameters:

x	value to be transformed.
v	order of bessel function (integer).

Authors:

Stephen L. Moshier. Copyright 1984, 1987, 2000
Documentation by Will Bateman

Returns the Bessel function of the first kinds for any order (v)

Several expansions are included: the ascending power series, the Hankel expansion, and two transitional expansions for large v. If v is not too large, it is reduced by recurrence to a region of best accuracy.

The transitional expansions give 12D accuracy for v > 500.

Accuracy:

Results for integer v are calculated with x varying from -125 to +125. Otherwise, x ranges from 0 to 125. Error criterion is absolute, except relative when |J()| > 1.

  arithmetic  v domain  x domain    # trials      peak       rms
     IEEE      0,125     0,125      100000      4.6e-15    2.2e-16
     IEEE     -125,0     0,125       40000      5.4e-11    3.7e-13
     IEEE      0,500     0,500       20000      4.4e-15    4.0e-16
  Integer v:
     IEEE    -125,125   -125,125      50000      3.5e-15    1.9e-16
 

Example:

#include <stdio.h>
#include <codecogs/maths/special/bessel/j/j.h>
 
int main()
{
  printf("\n  x      v=0.5    v=1.5    v=2.5    v=3.5    v=4.5");
  for(double x=0; x<6; x++)
  {
    printf("\nx=%.1lf",x);
    for(double v=0.5;v<=5;v++)
      printf(" %8.6lf", Maths::Special::Bessel::J::J(x,v));
  }
  return 0;
}

Output:

x      v=0.5    v=1.5    v=2.5    v=3.5    v=4.5
x=0.0 0.000000 0.000000 0.000000 0.000000 0.000000
x=1.0 0.671397 0.240298 0.049497 0.007186 0.000807
x=2.0 0.513016 0.491294 0.223925 0.068518 0.015887
x=3.0 0.065008 0.477718 0.412710 0.210132 0.077598
x=4.0 -0.301921 0.185286 0.440885 0.365820 0.199300
x=5.0 -0.342168 -0.169651 0.240377 0.410029 0.333663

References:

Cephes Math Library Release 2.8: June, 2000

Parameters:

x	input argument.
v	order of bessel function.

Authors:

Stephen L. Moshier. Copyright 1984, 1987, 2000
Documentation by Will Bateman