# Kelvin functions

From Encyclopedia of Mathematics

*Thomson functions*

The functions and , and , and , defined by

where the are the Hankel functions and the are the Bessel functions. When the index is omitted. The Kelvin functions form a fundamental system of solutions of the equation

which for turns into the Bessel equation.

The series representations are:

The asymptotic representations are:

where

These functions were introduced by W. Thomson (Lord Kelvin, [1]).

#### References

[1] | W. Thomson, "Mathematical and physical papers" , 3 , Cambridge Univ. Press (1980) pp. 492 |

[2] | E. Jahnke, F. Emde, F. Lösch, "Tafeln höheren Funktionen" , Teubner (1966) |

[3] | I.S. Gradshtein, I.M. Ryzhik, "Table of integrals, series and products" , Acad. Press (1973) (Translated from Russian) |

#### Comments

#### References

[a1] | M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1965) |

**How to Cite This Entry:**

Kelvin functions. A.B. Ivanov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Kelvin_functions&oldid=15392

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098