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BESSEL FUNCTION
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See also:BESSEL See also:FUNCTION , a certain mathematical relation between two variables. The Bessel function\ of See also:order m satisfies the See also:differential See also:equation _! + p j((p + (I —P z I u = o, and may be expressed dp2 as the See also:series `Dml 1- P2 2.2m + 2 /+ 2 2.4.2m+2.2m+4 ... ; the function of zero order is deduced by making In= o, and is See also:equivalent to the series 1-4 + ,v.4. &c. O. Schlomilch defines these functions as the coefficients of the See also:power of t in the expansion of exp Zp(t—t-'). The See also:symbol generally adopted to represent these functions is Jm (p) where m denotes the order of the function. These functions are named after See also:Friedrich Wilhelm Eessel, who in 1817 introduced them in an investigation on See also: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 See also:Daniel See also:Bernoulli obtained the function of zero order as a See also:solution to the problem of the oscillations of a See also:chain suspended at one end. This problem has been more fully discussed by See also:Sir A. G. Greenhill. In 1764 Leonhard See also:Euler employed the functions of both zero and integral orders in an See also:analysis into the vibrations of a stretched membrane; an investigation which has been considerably See also:developed by See also:Lord See also:Rayleigh, who has also shown (1878) that Bessel's functions are particular cases of See also:Laplace's functions. There is hardly a See also:branch of mathematical physics which is See also:independent of these functions. Of the many applications we may See also:notice:—Joseph See also:Fourier's (1824) investigation of the See also:motion of See also:heat in a solid See also:cylinder, a problem which, with the related one of the flow of See also:electricity, has been developed by W. E. See also:Weber, G. F. See also:Riemann and S. D. See also:Poisson; the flow of electromagnetic waves along wires (Sir J. J. Thom-son, H. See also:Hertz, O. Heaviside); the diffraction of See also:light (E. Lommel, Lord Rayleigh, Georg Wilhelm See also:Struve); the theory of See also:elasticity (A. E. Love, H. See also:Lamb, C. Chree, Lord Rayleigh); and to See also:hydrodynamics (Lord See also:Kelvin, Sir G. See also:Stokes). The remarkable connexion between Bessel's functions and spherical harmonics was established in 1868 by F. G. Mehler, who proved that a See also:simple relation existed between the function of zero order and the zonal See also:harmonic of order n. Heinrich Eduard See also: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 SPIP'RICAL HARMONICS and for tables see TABLE, MATHEMATICAL.
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