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MathsSpecialBesselK

K

Modified Bessel function, third kind, integer order.
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C++
Excel

K

 
doubleKdoublex
intv )
The modified Bessel function of the third kind (or Weber Function) is a linear combination of Bessel functions of the first and second kinds. These are called the Hankel functions of the first and second kinds, and are one of two types of functions known as Hankel functions. The other type of Hankel function is defined by a contour integral.

1/besselk-969.gif
+
Modified Bessel function of the 2nd/3rd kind
See also Maths/Special/Bessel/Hankel

This function returns modified Bessel function of the third kind of order n of the argument.

The range is partitioned into the two intervals [0,9.55] and (9.55, infinity). An ascending power series is used in the low range, and an asymptotic expansion in the high range.

Algorithm for Kn.

where p(m) is the psi function: p(1) = -EUL and

For large x, asymptotically, where

Accuracy:

<pre> Relative error: arithmetic domain # trials peak rms DEC 0,30 3000 1.3e-9 5.8e-11 IEEE 0,30 90000 1.8e-8 3.0e-10 </pre> Error is high only near the crossover point x = 9.55 between the two expansions used.

References:

Cephes Math Library Release 2.8: June, 2000

Example 1

#include <stdio.h>
#include <codecogs/maths/special/bessel/k/k.h>
 
int main()
{
printf("\n  x      v=0      v=1      v=2      v=3      v=4      v=5");
for(double x=1; x<7; x++)
{
  printf("\nx=%.1lf",x);
  for(int v=0;v<=5;v++)
    printf(" %8.6lf", Maths::Special::Bessel::K::K(x,v));
}
return 0;
}
Output:
x      v=0      v=1      v=2      v=3      v=4      v=5
x=1.0 0.421024 0.601907 1.624839 7.101263 44.232416 360.960590
x=2.0 0.113894 0.139866 0.253760 0.647385 2.195916 9.431049
x=3.0 0.034740 0.040156 0.061510 0.122170 0.305851 0.937774
x=4.0 0.011160 0.012483 0.017401 0.029885 0.062229 0.154343
x=5.0 0.003691 0.004045 0.005309 0.008292 0.015259 0.032706
x=6.0 0.001244 0.001344 0.001692 0.002472 0.004164 0.008024

Parameters

xinput argument
vorder

Authors

Stephen L. Moshier. Copyright 1984, 1987, 2000
Documentation by Nick Owens
Source Code

Source code is available when you buy a Commercial licence.

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