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.

See also 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 . Referenced
on Wolfram|Alpha Bessel Function
Cite this as:
Weisstein, Eric W. "Bessel Function."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/BesselFunction.html

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