## Modified Bessel Function of the Second Kind

The function which is one of the solutions to the Modified Bessel Differential Equation. The above plot shows for , 2, ..., 5. is closely related to the Modified Bessel Function of the First Kind and Hankel Function ,

 (1) (2) (3)

(Watson 1966, p. 185). A sum formula for is

 (4)
where is the Digamma Function (Abramowitz and Stegun 1972). An integral formula is

 (5)

which, for , simplifies to
 (6)

Other identities are
 (7)

for and

 (8) (9)

The modified Bessel function of the second kind is sometimes called the Basset Function.

References

Abramowitz, M. and Stegun, C. A. (Eds.). Modified Bessel Functions and .'' §9.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 374-377, 1972.

Arfken, G. Modified Bessel Functions, and .'' §11.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 610-616, 1985.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Modified Bessel Functions of Integral Order'' and Bessel Functions of Fractional Order, Airy Functions, Spherical Bessel Functions.'' §6.6 and 6.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 229-245, 1992.

Spanier, J. and Oldham, K. B. The Basset .'' Ch. 51 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 499-507, 1987.

Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.

© 1996-9 Eric W. Weisstein
1999-05-26