Kelvin Functions

Kelvin defined the Kelvin functions Bei and Ber according to

$$J_\nu(xe^{3\pi i/4})=\mathop{\rm ber}\nolimits _\nu(x)+i\mathop{\rm bei}\nolimits _\nu(x),$$ (1)

where $J_\nu(s)$ is a Bessel Function of the First Kind, and the functions Kei and Ker by

$$e^{-\nu\pi i/2}K_\nu(xe^{\pi i/4})=\mathop{\rm ker}\nolimits _\nu(x)+i\mathop{\rm kei}\nolimits _\nu(x),$$ (2)

where $K_\nu(x)$ is a Modified Bessel Function of the Second Kind. For the special case $\nu=0$,

$$J_0(i\sqrt{i}\,x)=J_0({\textstyle{1\over 2}}\sqrt{2}(i-1)x) ... ...\mathop{\rm ber}\nolimits (x)+i \mathop{\rm bei}\nolimits (x).$$ (3)

See also Bei, Ber, Kei, Ker

References

Abramowitz, M. and Stegun, C. 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.

Spanier, J. and Oldham, K. B. ``The Kelvin Functions.'' Ch. 55 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 543-554, 1987.