**BESSEL FUNCTION,** a certain mathematical relation between two variables. The *Bessel function of order m* satisfies the differential equation
*d* ^{2}*u**d*ρ^{2}＋1ρ *du**d*ρ＋(1 − *m*^{2}ρ^{2}) *u*＝0
and may be expressed as the series
ρ^{m}2^{m}.*m!* 1−ρ^{2}2.2*m*+2＋ρ^{4}2.4.2*m*+2.2*m*+4 . . .;
the function of *zero order* is deduced by making *m*＝0, and is equivalent to the series 1−ρ^{2}2^{2}＋ρ^{4}2^{2}.4^{2}, &c. O. Schlömilch defines these functions as the coefficients of the power of *t* in the expansion of exp 12ρ(*t*−*t* ^{−1}). The symbol generally adopted to represent these functions is J_{m}(ρ) where *m* denotes the order of the function. These functions are named after Friedrich Wilhelm Bessel, who in 1817 introduced them in an investigation on Kepler’s Problem. He discussed their properties and constructed tables for their evaluation. Although Bessel was the first to systematically treat of these functions, it is to be noted that in 1732 Daniel Bernoulli obtained the function of zero order as a solution to the problem of the oscillations of a chain suspended at one end. This problem has been more fully discussed by Sir A. G. Greenhill. In 1764 Leonhard Euler employed the functions of both zero and integral orders in an analysis into the vibrations of a stretched membrane; an investigation which has been considerably developed by Lord Rayleigh, who has also shown (1878) that Bessel’s functions are particular cases of Laplace’s functions. There is hardly a branch of mathematical physics which is independent of these functions. Of the many applications we may notice:—Joseph Fourier’s (1824) investigation of the motion of heat in a solid cylinder, a problem which, with the related one of the flow of electricity, has been developed by W. E. Weber, G. F. Riemann and S. D. Poisson; the flow of electromagnetic waves along wires (Sir J. J. Thomson, H. Hertz, O. Heaviside); the diffraction of light (E. Lömmel, Lord Rayleigh, Georg Wilhelm Struve); the theory of elasticity (A. E. Love, H. Lamb, C. Chree, Lord Rayleigh); and to hydrodynamics (Lord Kelvin, Sir G. Stokes).

The remarkable connexion between Bessel’s functions and spherical harmonics was established in 1868 by F. G. Mehler, who proved that a simple relation existed between the function of zero order and the zonal harmonic of order *n*. Heinrich Eduard Heine has shown that the functions of higher orders may be considered as limiting values of the associated functions; this relation was discussed independently, in 1878, by Lord Rayleigh.

For the mathematical investigation see Spherical Harmonics and for tables see Table, Mathematical.

See A. Gray and G. B. Matthews, *Treatise on Bessel’s Functions* (1895); *Encyclopädie der math. Wissenschaften*; F. W. Bessel, *Untersuchung des Teils der planetarischen Störungen* (1824).