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# K0

Modified Bessel function of the third kind, with zero order.
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Contents

C++
Excel

## Overview

Modified Bessel function, third kind, order zero.

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.

The Bessel K ,$\inline&space;K_v$, is defined by the following second order differential equation:

$x&space;\frac{\partial&space;y&space;(x)}{\partial&space;x}&space;+&space;x^{2}&space;\frac{\partial^{2}&space;y&space;(x)}{\partial&space;x^{2}}&space;-(x^{2})y&space;=0$

## K0

 doubleK0( double x )
Returns modified Bessel function of the third kind of order zero of the argument.

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

## Accuracy:

Tested at 2000 random points between 0 and 8. Peak absolute error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15. <pre> Relative error: domain # trials peak rms 0, 30 30000 1.2e-15 1.6e-16 </pre>

## Example:

#include <stdio.h>
#include <codecogs/maths/special/bessel/k/k0.h>

int main()
{
for(double x=1; x<7; x+=1)
{
double y=Maths::Special::Bessel::K::K0(x);
printf("\n K0(%.1lf)=%lf", x,y);
}
return 0;
}

## Output:

K0(1.0)=0.421024
K0(2.0)=0.113894
K0(3.0)=0.034740
K0(4.0)=0.011160
K0(5.0)=0.003691
K0(6.0)=0.001244

## References:

Cephes Math Library Release 2.8: June, 2000

### Parameters

 x input argument

### Authors

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

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

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## K0 Exp

 doubleK0_exp( double x )
Returns exponentially scaled modified Bessel function of the third kind of order zero of the argument.

## Accuracy:

<pre> Relative error: domain # trials peak rms 0, 30 30000 1.4e-15 1.4e-16 </pre>

## Example:

#include <stdio.h>
#include <codecogs/maths/special/bessel/k/k0.h>

int main()
{
for(double x=1; x<7; x+=1)
{
double y=Maths::Special::Bessel::K::K0_exp(x);
printf("\n K0_exp(%.1lf)=%lf", x,y);
}
return 0;
}

## Output:

K0_exp(1.0)=1.144463
K0_exp(2.0)=0.841568
K0_exp(3.0)=0.697762
K0_exp(4.0)=0.609298
K0_exp(5.0)=0.547808
K0_exp(6.0)=0.501863

## References:

Cephes Math Library Release 2.8: June, 2000

### Parameters

 x input argument

### Authors

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