---
tag: 77BS2B0A6Q
kind: concept
name: Kelvin functions
msc: [33C10]
sources:
  - url: https://en.wikipedia.org/wiki/Kelvin_functions
same_as:
  wikidata: Q4493413
  wikipedia: Kelvin functions
---

## Summary

The Kelvin functions $\operatorname{ber}_\nu$, $\operatorname{bei}_\nu$, $\operatorname{ker}_\nu$ and $\operatorname{kei}_\nu$ are the real and imaginary parts of Bessel functions evaluated along the ray $\arg z = \frac{3\pi}{4}$ or $\frac{\pi}{4}$.
They are real functions of a positive real variable.
They describe alternating current in a round wire, where the current crowds toward the surface (the skin effect).

## Explanations

### L1 human

Some physical problems lead to Bessel's equation with a complex coefficient, such as $x^2 w'' + x w' - i x^2 w = 0$.
Its solutions are complex numbers, and one splits them into real and imaginary parts.
These parts have names: $\operatorname{ber}$ and $\operatorname{bei}$ for the solution that is finite at $0$, and $\operatorname{ker}$ and $\operatorname{kei}$ for the one that decays at infinity.
For example, $\operatorname{ber}_0(x) = 1 - \frac{(x/2)^4}{(2!)^2} + \frac{(x/2)^8}{(4!)^2} - \cdots$.