TOPICS
Search

Kei

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
Kei5

The kei_nu(z) function is defined as the imaginary part of

e^(-nupii/2)K_nu(ze^(pii/4))=ker_nu(z)+ikei_nu(z),
(1)

where K_nu(z) is a modified Bessel function of the second kind. Therefore,

kei_nu(z)=I[e^(-nupii/2)K_nu(ze^(pii/4))],
(2)

where I[z] is the imaginary part.

It is implemented as KelvinKei[nu, z].

ker_n(z) has a complicated series given by Abramowitz and Stegun (1972, p. 380).

Kei
KeiContours

The special case nu=0 is commonly denoted kei_0(z)=kei(z) and has the plot shown above.

kei(z) 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),
(3)

where psi(z) 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 ber_nu(x), beinu(x), ker_nu(x) and kei_nu(x)." §1.7 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 29-30, 1990.

Referenced on Wolfram|Alpha

Kei

Cite this as:

Weisstein, Eric W. "Kei." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Kei.html

Subject classifications