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).

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. 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.