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# Bessel Function

A function defined by the recurrence relations

 (1)

and

 (2)

The Bessel functions are more frequently defined as solutions to the differential equation

 (3)

There are two classes of solution, called the Bessel function of the first kind and Bessel function of the second kind . (A Bessel function of the third kind, more commonly called a Hankel function, is a special combination of the first and second kinds.) Several related functions are also defined by slightly modifying the defining equations.

Bessel Function of the First Kind, Bessel Function of the Second Kind, Cylinder Function, Hemicylindrical Function, Modified Bessel Function of the First Kind, Modified Bessel Function of the Second Kind, Spherical Bessel Function of the First Kind, Spherical Bessel Function of the Second Kind

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## References

Abramowitz, M. and Stegun, I. A. (Eds.). "Bessel Functions of Integer Order," "Bessel Functions of Fractional Order," and "Integrals of Bessel Functions." Chs. 9-11 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 355-389, 435-456, and 480-491, 1972.Adamchik, V. "The Evaluation of Integrals of Bessel Functions via -Function Identities." J. Comput. Appl. Math. 64, 283-290, 1995.Arfken, G. "Bessel Functions." Ch. 11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 573-636, 1985.Bickley, W. G. Bessel Functions and Formulae. Cambridge, England: Cambridge University Press, 1957.Bowman, F. Introduction to Bessel Functions. New York: Dover, 1958.Byerly, W. E. "Cylindrical Harmonics (Bessel's Functions)." Ch. 7 in An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 219-237, 1959.Gray, A. and Mathews, G. B. A Treatise on Bessel Functions and Their Applications to Physics, 2nd ed. New York: Dover, 1966.Luke, Y. L. Integrals of Bessel Functions. New York: McGraw-Hill, 1962.McLachlan, N. W. Bessel Functions for Engineers, 2nd ed. with corrections. Oxford, England: Clarendon Press, 1961.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Bessel Functions of Integral Order" and "Bessel Functions of Fractional Order, Airy Functions, Spherical Bessel Functions." §6.5 and 6.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 223-229 and 234-245, 1992.Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.Weisstein, E. W. "Books about Bessel Functions." http://www.ericweisstein.com/encyclopedias/books/BesselFunctions.html.

Bessel Function

## Cite this as:

Weisstein, Eric W. "Bessel Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BesselFunction.html