kelvin-package | R Documentation |
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.
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
.
Andrew Barbour <andy.barbour@gmail.com>
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
Useful links:
Fundamental solution: Beir
Equivalent solution: Keir
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.