Bessel Function

A function $Z_n(x)$ defined by the Recurrence Relations

$$Z_{n+1}+Z_{n-1} = {2n\over x} Z_n$$

and

$$Z_{n+1}-Z_{n-1} = -2 {dZ_n\over dx}.$$

The Bessel functions are more frequently defined as solutions to the Differential Equation

$$x^2{d^2y\over dx^2} + x {dy\over dx} + (x^2-n^2)y = 0.$$

There are two classes of solution, called the Bessel Function of the First Kind $J_n(x)$ and Bessel Function of the Second Kind $Y_n(x)$. (A Bessel Function of the Third Kind 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, Bessel Function of the Third 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

References

Bessel Functions

Abramowitz, M. and Stegun, C. 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.

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.

Gray, A. and Matthews, 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.