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Modified Bessel function, third kind, order one.

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Interface

#include <codecogs/maths/special/bessel/k/k1.h>

using namespace Maths::Special::Bessel::K;

double K1(double x)
double K1_exp(double x )

Overview

Modified Bessel function, thirdkind, order one.

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.

Let $x$ be a complex variable of $\setminus \{0,\infty\}$ and let $\nu$ denote a parameter (independent of $x$ ). The function Bessel K (noted $K_{\nu}$ ) is defined by the following second order differential equation:

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

(1)

Although $0$ is a singularity, the initial conditions can be given by:

$\left [ x^{(-\nu)}\right ] K_{\nu} (x) =2^{(\nu - 1)} \Gamma (\nu)$

(2)

$\left[ x^{\nu} \right ] K_{\nu} (x) =-\frac{\pi}{\Gamma (\mu + 1) \sin (\mu \pi) 2^{(\mu + 1)}}$

(3)

FUNCTION

K1

Computes the modified Bessel function of the third kind of order one of the argument.

The range is partitioned into the two intervals [0,2] and [2, infinity]. Chebyshev polynomial expansions are employed in each interval.

Accuracy:

<pre> Relative error: domain # trials peak rms 0, 30 3300 8.9e-17 2.2e-17 0, 30 30000 1.2e-15 1.6e-16 </pre>

Example:

#include <stdio.h>
#include <codecogs/maths/special/bessel/k/k1.h>

int main()
{
for(double x=1; x<7; x+=1)
{
  double y=Maths::Special::Bessel::K::K1(x);
  printf("\n K1(%.1lf)=%lf", x,y);
}
return 0;
}

Output:

K1(1.0)=0.601907
K1(2.0)=0.139866
K1(3.0)=0.040156
K1(4.0)=0.012483
K1(5.0)=0.004045
K1(6.0)=0.001344

References

Cephes Math Library Release 2.8: June, 2000

Parameters

input argument

Author

Documentation by Nick Owens

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