Bei

$\begin{figure}\begin{center}\BoxedEPSF{Bei.epsf scaled 800}\end{center}\end{figure}$

$\begin{figure}\begin{center}\BoxedEPSF{BeiReIm.epsf scaled 800}\end{center}\end{figure}$

The Imaginary Part of

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

(1)

The special case $\nu=0$ gives

$\begin{displaymath} J_0(i\sqrt{i}\,x) \equiv \mathop{\rm ber}\nolimits (x)+i \mathop{\rm bei}\nolimits (x), \end{displaymath}$

(2)

where

is the zeroth order Bessel Function of the First Kind.

$\begin{displaymath} \mathop{\rm bei}\nolimits (x) \equiv \sum_{n=0}^\infty {(-1)^n ({\textstyle{1\over 2}}x)^{2+4n}\over [(2n+1)!]^2}. \end{displaymath}$

(3)

See also Ber, Bessel Function, Kei, Kelvin Functions, 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.