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# K

Modified Bessel function, third kind, integer order.
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Excel

## K

 doubleK( double x int v )
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.

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

 x input argument v order

### 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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Last Modified: 18 Oct 07 @ 17:07     Page Rendered: 2022-03-14 15:05:02