Kelvin Functions
Kelvin defined the Kelvin functions bei and ber according to
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
is a Bessel
function of the first kind and 
is a modified Bessel function of the
first kind. These functions satisfy the Kelvin
differential equation.
Similarly, the functions kei and ker by
 |
(5)
|
where 
is a modified
Bessel function of the second kind. For the special case 
,
 |  |  |
(6)
|
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(7)
|
Abramowitz, M. and Stegun, I. A. (Eds.). "Kelvin Functions." §9.9 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 379-381, 1972.
Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Kelvin Functions 
, 
, 
and 
." §1.7 in Integrals
and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach,
pp. 29-30, 1990.
Spanier, J. and Oldham, K. B. "The Kelvin Functions." Ch. 55 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 543-554, 1987.
Weisstein, Eric W. "Kelvin Functions." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KelvinFunctions.html
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