The 
function is defined as the imaginary part of
 |
(1) |
where 
is a modified
Bessel function of the second kind. Therefore,
![kei_nu(z)=I[e^(-nupii/2)K_nu(ze^(pii/4))],](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FKei%2FNumberedEquation2.svg){kind=small} |
(2) |
where ![I[z]](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FKei%2FInline3.svg){kind=small}
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
![kei(z)=-ln(1/2z)bei(z)-1/4piber(z)
+sum_(k=0)^infty(-1)^k(psi(2k+2))/([(2k+1)!]^2)(1/4z^2)^(2k+1),](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FKei%2FNumberedEquation3.svg) |
(3) |
where 
is the digamma
function (Abramowitz and Stegun 1972, p. 380).
See also
Bei, Ber, Ker, Kelvin Functions
Explore with Wolfram|Alpha
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.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Kei." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Kei.html