# Kelvin

Kelvin functions in R

This project is maintained by abarbour

# K E L V I N

Calculation of solutions to the Kelvin Differential Equation using Bessel functions. This package is written for `R`, and uses primarily `Bessel::BesselJ` and `Bessel::BesselK` to calculate the appropriate Kelvin Functions.

## Background information

The following is taken from http://mathworld.wolfram.com/KelvinFunctions.html

Kelvin defined the Kelvin functions bei and ber according to

``````ber_v(x) + i*bei_v(x)
=   J_v(x*exp(2*pi*i/4))
=   exp(v*pi*i)*J_v(x*exp(-pi*i/4))
=   exp(v*pi*i/2)*I_v(x*exp(pi*i/4))
=   exp(3*v*pi*i/2)*I_v(x*exp(-3*pi*i/4))
``````

where J_v(x) is a Bessel function of the first kind and I_v(x)
is a modified Bessel function of the first kind. These functions satisfy the Kelvin differential equation.

Similarly, the functions kei and ker by

``````ker_v(x) + i*kei_v(x) = exp(-v*pi*i/2)*K_v(x*exp(pi*i/4))
``````

where K_v(x) is a modified Bessel function of the second kind. For the special case v=0,

``````J_0(i*sqrt(i)*x)
=   J_0(sqrt(2)*(i-1)*x/2)
=   ber(x) + i*bei(x)
``````