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.
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)