Bessel Function

BESSEL FUNCTION, in mathematics, a solution of Bes sel's differential equation The designation cylinder function, also used by many writers, has its source in the use of these functions to express solutions of such physical problems as the flow of heat or of electricity in a solid circular cylinder. They first appear in this connection in the work of J. B. J. Fourier (La Theorie analytique de la Chaleur, 1822, and a memoir of I 8 I 1, published in 1824). They had, how ever, been previously employed in other physical and astronomical problems. In 1732 Daniel Bernoulli investigated the transverse oscillations of a heavy chain suspended at one end, and expressed the horizontal displacement in terms of a Bessel function. Leon hard Euler found, in 1764, that these functions could be used to evaluate the displacement of a point on a vibrating stretched membrane. Their first use in celestial mechanics is in Lagrange's memoir of 177o on Kepler's Problem concerning the motion of a planet about the sun. They were also employed by other writers of the earlier classical period of mathematics (Poisson, 1823, Laplace, Mecanique Celeste, 1827). F. W. Bessel, after whom these functions are named, made a systematic investigation of their properties in his memoir of 1824 on Kepler's Problem, fol lowing his earlier paper of 1817 on the same subject.

The importance of Bessel functions in mathematical physics

and astronomy is indicated by the foregoing examples of their applications, to which we may add more modern solutions of problems in wave-theory, elasticity, hydrodynamics, and optics. The bibliography of G. N. Watson (A Treatise on the Theory of Bessel Functions, Cambridge University Press, 192 2) contains a list of over seven hundred papers on these functions and their applications.

In modern treatises Bessel functions are considered as depend ing both upon x and n, which may have complex as well as real values. The value of n is called the order of the function. Three kinds are distinguished. The Bessel function of the first kind, designated by J„ (x), is given by the infinite series xn( ? n L 1 — --{- — 1, 2 I' n I J ? 4 ? 21a 4) where T (1a-}-1) is the gamma function, which is equal to product of all integers from i to 11 when n is a positive Functions of the second kind, designated by (x) and K n are variously defined, but are expressible as linear of functions J,, (x) and J-,, (x), or limits of such while functions of the third kind, H, (x), are of the form J„ (x) (x).

These functions are expressible as limiting cases of spherical

harmonics. They also appear as coefficients in series expansions of certain elementary functions. Extensive studies have been made of the problem of expressing an arbitrary function in terms of infinite series or of definite integrals in terms of Bessel func tions. In this connection it has been necessary to determine the zeros of Bessel functions. Asymptotic expansions have also fur nished an extensive field of investigation.

Many tables of these functions have been published. Among

the best and most extensive are those of Watson in the work already quoted.

See, in addition to the Treatise of Watson, Encyclopadie der math. Wissenschaften (II.,A,io,VIII.), and the corresponding section of the French Encyclopedie; A. Gray and G. B. Mathews, A Treatise on Bessel's Functions (1895) ; N. Nielsen, Handbuch der Theorie der Cylinderfunktionen (Leipzig, 1904) • (D. R. C.)