Description Details Author(s) References See Also

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} - ≤ft(i x^2 + ν^2\right) y = 0*

and has a suite of complex solutions. One set of solutions,
*\mathcal{B}_ν*, is defined in the following manner:

*\mathcal{B}_ν \equiv \mathrm{Ber}_ν (x) + i \mathrm{Bei}_ν (x)*

*= J_ν ≤ft(x \cdot \exp(3 π i / 4)\right) *

*= \exp(ν π i) \cdot J_ν ≤ft(x \cdot \exp(-π i / 4)\right)*

*= \exp(ν π i / 2) \cdot I_ν ≤ft(x \cdot \exp(π i / 4)\right)*

*= \exp(3 ν π i / 2) \cdot I_ν ≤ft(x \cdot \exp(-3 π i / 4)\right)*

where
*J_ν* is a Bessel function of the first kind, and
*I_ν* is a *modified* Bessel function of the first kind.

Similarly, the complementary solutions, *\mathcal{K}_ν*,
are defined as

*\mathcal{K}_ν \equiv \mathrm{Ker}_ν (x) + i \mathrm{Kei}_ν (x)*

*= \exp(- ν π i / 2) \cdot K_ν ≤ft(x \cdot \exp(π i / 4)\right)*

where *K_ν* is a *modified* Bessel function of the second kind.

The relationships between *y* in the differential equation, and
the solutions *\mathcal{B}_ν* and *\mathcal{K}_ν* are as follows

* y = \mathrm{Ber}_ν (x) + i \mathrm{Bei}_ν (x)*

* = \mathrm{Ber}_{-ν} (x) + i \mathrm{Bei}_{-ν} (x)*

* = \mathrm{Ker}_ν (x) + i \mathrm{Kei}_ν (x)*

* = \mathrm{Ker}_{-ν} (x) + i \mathrm{Kei}_{-ν} (x)*

In the case where *ν=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 ≤ft(i √{i} \cdot x\right)*

* = J_0 ≤ft(√{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}_ν* and *\mathcal{K}_ν*.

Andrew Barbour <[email protected]>

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`

kelvin documentation built on May 2, 2019, 3:38 p.m.

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