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Kelvin Functions

Kelvin defined the Kelvin functions bei and ber according to

ber_nu(x)+ibei_nu(x)=J_nu(xe^(3pii/4))
(1)
=e^(nupii)J_nu(xe^(-pii/4)),
(2)
=e^(nupii/2)I_nu(xe^(pii/4))
(3)
=e^(3nupii/2)I_nu(xe^(-3pii/4)),
(4)

where J_nu(x) is a Bessel function of the first kind and I_nu(x) is a modified Bessel function of the first kind. These functions satisfy the Kelvin differential equation.

Similarly, the functions kei and ker by

ker_nu(x)+ikei_nu(x)=e^(-nupii/2)K_nu(xe^(pii/4)),
(5)

where K_nu(x) is a modified Bessel function of the second kind. For the special case nu=0,

J_0(isqrt(i)x)=J_0(1/2sqrt(2)(i-1)x)
(6)
=ber(x)+ibei(x).
(7)

See also

Bei, Ber, Kei, Kelvin Differential Equation, 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.

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Kelvin Functions

Cite this as:

Weisstein, Eric W. "Kelvin Functions." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KelvinFunctions.html

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