README.md

In abarbour/kelvin: Calculate Solutions to the Kelvin Differential Equation using Bessel Functions

kelvin

Solutions to the the Kelvin differential equation in R

by Andrew J Barbour

Calculation of solutions to the Kelvin Differential Equation using Bessel functions namely BesselJ and BesselK from the Bessel package.

Background information

The following is taken from Wolfram:

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)

abarbour/kelvin documentation built on June 10, 2025, 1:05 p.m.