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Modified Bessel function of the first kind of integer order.

View versions (1)

Interface

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

using namespace Maths::Special::Bessel::I;

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

Overview

These function return solutions to the Modified Bessel Function of the first kind.

The differential equation

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

(1)

where v is a real constant, is called the modified Bessel's equation, with the solution known as the modified Bessel function, with solutions: $I_v(z)$ and $I_{-v}(z)$ , and $K_v(z)$ where

$I_{v}(z) = \frac{z}{2}^v \sum_{k=0}^{\infty} \frac{\left ( \frac{1}{4}z^2 \right )^k}{k! \Gamma(v+k+1)}$

(2)

where $\Gamma(n)$ is the gamma function.

A second solution, which is independent of $I_{v}(z)$ , is known as the modified Bessel function of the kind K

References

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

FUNCTION

I

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

The function is defined as $I_1(x) = -i H_1(ix)$

The range is partitioned into the two intervals [0,8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval.

Accuracy:

<pre> Relative error: arithmetic domain # trials peak rms DEC 0, 30 3400 1.2e-16 2.3e-17 IEEE 0, 30 30000 1.9e-15 2.1e-16</pre>

References

Cephes Math Library Release 2.8: June, 2000

Parameters

value to be transformed.

order of bessel function.

Author

Documentation by Will Bateman (August 2005)

Interactive Calculator

Result –

FUNCTION

I

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

The function is defined as $I_1(x) = -i J_1(ix)$

The range is partitioned into the two intervals [0,8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval.

Accuracy:

<pre> Relative error: arithmetic domain # trials peak rms DEC 0, 30 3400 1.2e-16 2.3e-17 IEEE 0, 30 30000 1.9e-15 2.1e-16</pre>

Example:

#include <codecogs/maths/special/bessel/i/i.h>
#include <stdio.h>

int main()
{
  using namespace Maths::Special::Bessel::I;
  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", I(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 1.266066 0.565159 0.135748 0.022168 0.002737 0.000271
x=2.0 2.279585 1.590637 0.688948 0.212740 0.050729 0.009826
x=3.0 4.880793 3.953370 2.245212 0.959754 0.325705 0.091206
x=4.0 11.301922 9.759465 6.422189 3.337276 1.416276 0.504724
x=5.0 27.239872 24.335642 17.505615 10.331150 5.108235 2.157975

References

Cephes Math Library Release 2.8: June, 2000

Parameters

input argument.

order of bessel function.

Author

Documentation by Will Bateman (August 2005)

Interactive Calculator

Result –