Kei
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
"Kelvin Functions." §9.9 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
25-3-2019
2425
Kei
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))],](http://mathworld.wolfram.com/images/equations/Kei/NumberedEquation2.gif) |
(2)
|
where ![I[z]](http://mathworld.wolfram.com/images/equations/Kei/Inline3.gif)
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),](http://mathworld.wolfram.com/images/equations/Kei/NumberedEquation3.gif) |
(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.
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