The function is defined as the imaginary part of

(1)

where is a modified
Bessel function of the second kind. Therefore,

(2)

where is the imaginary
part.
It is implemented as KelvinKei[nu,
z].
has a complicated series given
by Abramowitz and Stegun (1972, p. 380).
The special case
is commonly denoted
and has the plot shown above.
has the series expansion

(3)

where is the digamma
function (Abramowitz and Stegun 1972, p. 380).
See also
Bei,
Ber,
Ker,
Kelvin Functions
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References
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. 379381, 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. 2930, 1990.Referenced on WolframAlpha
Kei
Cite this as:
Weisstein, Eric W. "Kei." From MathWorldA
Wolfram Web Resource. https://mathworld.wolfram.com/Kei.html
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