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Bei

Bei5

The bei_nu(z) function is defined for real nu and positive real z through the equation

J_nu(ze^(3pii/4))=ber_nu(z)+ibei_nu(z),
(1)

where J_nu(z) is a Bessel function of the first kind, so

bei_nu(z)=Im[J_nu(ze^(3pii/4))],
(2)

where Im[z] is the imaginary part.

It is implemented in the Wolfram Language as KelvinBei[nu, z].

bei_nu(z) has the series expansion

bei_nu(x)=(1/2x)^nusum_(k=0)^infty(sin[(3/4nu+1/2k)pi])/(k!Gamma(nu+k+1))(1/4x^2)^k,
(3)

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

bei_nu(x)=-1/2ie^(-3piinu/4)x^nu[(-1)^(1/4)x]^(-nu)×[e^(3piinu/2)I_nu((-1)^(1/4)x)-J_nu((-1)^(1/4)x)],
(4)

where I_nu(z) is a modified Bessel function of the first kind.

Bei
BeiReIm
BeiContours

The special case nu=0, commonly denoted bei(z), corresponds to

J_0(isqrt(i)z)=ber(z)+ibei(z),
(5)

where J_0(x) is the zeroth order Bessel function of the first kind. The function bei_0(z)=bei(z) has the series expansion

bei(z)=sum_(n=0)^infty((-1)^n(1/2z)^(2+4n))/([(2n+1)!]^2).
(6)

Closed forms include

bei_0(z)=1/2i[J_0((-1)^(1/4)z)-I_0((-1)^(1/4)z)]
(7)
=1/2i[I_0((-1)^(3/4)z)-I_0((-1)^(1/4)z)].
(8)

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. 379-381, 1972.Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Kelvin Functions ber_nu(x), beinu(x), ker_nu(x) and kei_nu(x)." §1.7 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 29-30, 1990.Spanier, J. and Oldham, K. B. "The Kelvin Functions." Ch. 55 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 543-554, 1987.

Referenced on Wolfram|Alpha

Bei

Cite this as:

Weisstein, Eric W. "Bei." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Bei.html

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