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MathsSpecialBesselI

i

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Modified Bessel function of the first kind of integer order.
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Overview

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

The differential equation 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

where \Gamma(n) is the gamma function.

1/besselI-969.gif
+
The Bessel function at varying orders
A second solution, which is independent of I_{v}(z), is known as the modified Bessel function of the kind Maths/Special/Bessel/K/K

References:

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

I

 
doubleIdoublex
intv )[inline]
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

xvalue to be transformed.
vorder of bessel function.

Authors

Stephen L. Moshier. Copyright 1984, 1987, 2000
Documentation by Will Bateman (August 2005)
Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.


I

 
doubleIdoublex
doublev )
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

xinput argument.
vorder of bessel function.

Authors

Stephen L. Moshier. Copyright 1984, 1987, 2000
Documentation by Will Bateman (August 2005)
Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.