Kelvin functions

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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:


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


[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)



[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:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098