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

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

The Real Part of

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
J_0(i\sqrt{i}\,x) \equiv \mathop{\rm ber}\nolimits (x)+i \mathop{\rm bei}\nolimits (x),
\end{displaymath} (2)

where $J_0(x)$ is the zeroth order Bessel Function of the First Kind.
\mathop{\rm ber}\nolimits (x) \equiv \sum_{n=0}^\infty {(-1)^n ({\textstyle{1\over 2}}x)^{4n}\over [(2n)!]^2}.
\end{displaymath} (3)

See also Bei, Bessel Function, Kei, Kelvin Functions, Ker


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.

© 1996-9 Eric W. Weisstein