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Ker

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The symbol ker has at least two different meanings in mathematics. It can refer to a special function related to Bessel functions, or (written either with a capital or lower-case "K"), it can denote a kernel.

Ker5

The ker_nu(z) function is defined as the real part of

e^(-nupii/2)K_nu(ze^(pii/4))=ker_nu(z)+ikei_nu(z),
(1)

where K_nu(z) is a modified Bessel function of the second kind. Therefore

ker_nu(z)=R[e^(-nupii/2)K_nu(ze^(pii/4))],
(2)

where R[z] is the real part.

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

ker_n(z) has a complicated series given by Abramowitz and Stegun (1972, p. 379).

KerKerContours

850

The special case nu=0 is commonly denoted ker_0(z)=ker(z) and has the plot shown above. ker(z) has the series expansion

ker(x)=-ln(1/2x)ber(x)+1/4pibei(x) +sum_(k=0)^infty(-1)^k(psi(2k+1))/([(2k)!]^2)(1/4x^2)^(2k),
(3)

where psi(z) is the digamma function (Abramowitz and Stegun 1972, p. 380).

"ker" is also an abbreviation for "group kernel" of a group homomorphism.

See also

Bei, Ber, Group Kernel, Kei, Kelvin Functions

Portions of this entry contributed by Margherita Barile

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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.

Referenced on Wolfram|Alpha

Ker

Cite this as:

Barile, Margherita and Weisstein, Eric W. "Ker." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Ker.html

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