The functions here use Bessel functions to calculate the analytic solutions to the Kelvin differential equation, namely the fundamental (Be) and equivalent (Ke) complex functions.
Andrew Barbour <andy.barbour@gmail.com>
The complex second-order ordinary differential equation, known as the Kelvin differential equation, is defined as $$x^2 \ddot{y} + x \dot{y} - \left(i x^2 + \nu^2\right) y = 0$$ and has a suite of complex solutions. One set of solutions, \(\mathcal{B}_\nu\), is defined in the following manner: $$\mathcal{B}_\nu \equiv \mathrm{Ber}_\nu (x) + i \mathrm{Bei}_\nu (x)$$ $$= J_\nu \left(x \cdot \exp(3 \pi i / 4)\right) $$ $$= \exp(\nu \pi i) \cdot J_\nu \left(x \cdot \exp(-\pi i / 4)\right)$$ $$= \exp(\nu \pi i / 2) \cdot I_\nu \left(x \cdot \exp(\pi i / 4)\right)$$ $$= \exp(3 \nu \pi i / 2) \cdot I_\nu \left(x \cdot \exp(-3 \pi i / 4)\right)$$ where \(J_\nu\) is a Bessel function of the first kind, and \(I_\nu\) is a modified Bessel function of the first kind.
Similarly, the complementary solutions, \(\mathcal{K}_\nu\), are defined as $$\mathcal{K}_\nu \equiv \mathrm{Ker}_\nu (x) + i \mathrm{Kei}_\nu (x)$$ $$= \exp(- \nu \pi i / 2) \cdot K_\nu \left(x \cdot \exp(\pi i / 4)\right)$$ where \(K_\nu\) is a modified Bessel function of the second kind.
The relationships between \(y\) in the differential equation, and the solutions \(\mathcal{B}_\nu\) and \(\mathcal{K}_\nu\) are as follows $$ y = \mathrm{Ber}_\nu (x) + i \mathrm{Bei}_\nu (x)$$ $$ = \mathrm{Ber}_{-\nu} (x) + i \mathrm{Bei}_{-\nu} (x)$$ $$ = \mathrm{Ker}_\nu (x) + i \mathrm{Kei}_\nu (x)$$ $$ = \mathrm{Ker}_{-\nu} (x) + i \mathrm{Kei}_{-\nu} (x)$$
In the case where \(\nu=0\), the differential equation reduces to $$x^2 \ddot{y} + x \dot{y} - i x^2y = 0$$ which has the set of solutions: $$ J_0 \left(i \sqrt{i} \cdot x\right)$$ $$ = J_0 \left(\sqrt{2} \cdot (i-1) \cdot x / 2\right)$$ $$ = \mathrm{Ber}_0 (x) + i \mathrm{Bei}_0 (x) \equiv \mathcal{B}_0$$
This package has functions to calculate \(\mathcal{B}_\nu\) and \(\mathcal{K}_\nu\).
Abramowitz, M. and Stegun, I. A. (Eds.). "Kelvin Functions." \(\S 9.9\) in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 379-381, 1972.
Kelvin functions: http://mathworld.wolfram.com/KelvinFunctions.html
Bessel functions: http://mathworld.wolfram.com/BesselFunction.html
Fundamental solution: Beir
Equivalent solution: Keir