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Maths › Special › Bessel › K ›

K0

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

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Interface
K0
K0 Exp
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C++

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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 , $\inline K_v$ , is defined by the following second order differential equation:

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

(2)

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

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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.