CodeCogs

Bessel function of the first kind of integer order.

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Interface

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

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

inline double J(double x, int v)
double J(double x, double v)

Overview

These function return solutions to the Bessel Function of the first kind. They are defined as solutions to the Bessel differential equation

$x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + (x^2 - v^2)y = 0$

(1)

which are both nonsingular at the origin. They are are sometimes called cylinder functions or cylindrical harmonics.

References

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

FUNCTION

J

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:

<pre> 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 </pre>

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

References

Cephes Math Library Release 2.8: June, 2000

Parameters

value to be transformed.

order of bessel function (integer).

Author

Documentation by Will Bateman

Interactive Calculator

Result –

FUNCTION

J

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. <pre> 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 </pre>

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

input argument.

order of bessel function.

Author

Documentation by Will Bateman

Interactive Calculator

Result –