The function is defined through
the equation

(1)

where is a Bessel
function of the first kind, so

(2)

where is the imaginary
part.
It is implemented in the Wolfram Language as KelvinBei[nu,
z].
has the series expansion

(3)

where is the gamma
function (Abramowitz and Stegun 1972, p. 379), which can be written in closed
form as

(4)

where is a modified
Bessel function of the first kind.
The special case ,
commonly denoted ,
corresponds to

(5)

where is the zeroth order Bessel
function of the first kind. The function has the series expansion

(6)

Closed forms include
See also
Ber,
Bessel Function,
Kei,
Kelvin
Functions,
Ker
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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. 379381, 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. 2930, 1990.Spanier, J. and Oldham, K. B. "The Kelvin
Functions." Ch. 55 in An
Atlas of Functions. Washington, DC: Hemisphere, pp. 543554, 1987.Referenced
on WolframAlpha
Bei
Cite this as:
Weisstein, Eric W. "Bei." From MathWorldA
Wolfram Web Resource. https://mathworld.wolfram.com/Bei.html
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