Bessel function


Bessel function
one of several transcendental functions, usually represented as power series, that are solutions to a group of related differential equations.
[1870-75; named after F. W. BESSEL]

* * *

also called  Cylinder Function,  

      any of a set of mathematical functions systematically derived around 1817 by the German astronomer Friedrich Wilhelm Bessel during an investigation of solutions of one of Kepler's equations of planetary motion. Particular functions of the set had been formulated earlier by the Swiss mathematicians Daniel Bernoulli, who studied the oscillations of a chain suspended by one end, and Leonhard Euler, who analyzed the vibrations of a stretched membrane.

      After Bessel published his findings, other scientists found that the functions appeared in mathematical descriptions of many physical phenomena, including the flow of heat or electricity in a solid cylinder, the propagation of electromagnetic waves along wires, the diffraction of light, the motions of fluids, and the deformations of elastic bodies. One of these investigators, Lord Rayleigh, also placed the Bessel functions in a larger context by showing that they arise in the solution of Laplace's equation (q.v.) when the latter is formulated in cylindrical (rather than Cartesian or spherical) coordinates.

      Specifically, a Bessel function is a solution of the differential equation

      which is called Bessel's equation. For integral values of n, the Bessel functions are

 The graph of J0(x) looks like that of a damped cosine curve, and that of J1(x) looks like that of a damped sine curve (see Graph—>).

      Certain physical problems lead to differential equations analogous to Bessel's equation; their solutions take the form of combinations of Bessel functions and are called Bessel functions of the second or third kind.

* * *


Universalium. 2010.

Look at other dictionaries:

  • Bessel function — In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel s differential equation: for an arbitrary real or complex number α (the order of the …   Wikipedia

  • Bessel function — noun Any of a class of functions that are solutions to a particular form of differential equation (a Bessel equation) and are typically used to describe waves in a cylindrically symmetric system. Syn: cylinder function See Also: Bessel function… …   Wiktionary

  • Bessel function — Beselio funkcija statusas T sritis fizika atitikmenys: angl. Bessel function vok. Bessel Funktion, f rus. функция Бесселя, f pranc. fonction de Bessel, f …   Fizikos terminų žodynas

  • bessel function — ˈbesəl noun Usage: usually capitalized B Etymology: after Friedrich W. Bessel died 1846 Prussian astronomer : one of a class of transcendental functions expressible as infinite series and occurring in the solution of the differential equation x²… …   Useful english dictionary

  • Bessel function — noun Etymology: Friedrich W. Bessel died 1846 Prussian astronomer Date: 1872 one of a class of transcendental functions expressible as infinite series and occurring in the solution of the differential equation x² (d²y/dx²) + x (dy/dx) = (n² x²)y …   New Collegiate Dictionary

  • Bessel — may refer to: * Bessel beam * Bessel ellipsoid * Bessel function in mathematics * Bessel s inequality in mathematics * Bessel filter, a linear filter often used in audio crossover systems * Bessel (crater), a small lunar crater * Bessel transform …   Wikipedia

  • bessel — equation, bessel function, or bessel method (named for the German 19thcentury astronomer Friedrich Wilhelm Bessel) …   Eponyms, nicknames, and geographical games

  • Bessel-Clifford function — In mathematical analysis, the Bessel Clifford function is an entire function of two complex variables which can be used to provide an alternative development of the theory of Bessel functions. If :pi(x) = frac{1}{Pi(x)} = frac{1}{Gamma(x+1)}is… …   Wikipedia

  • Bessel polynomials — In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions. The definition favored by mathematicians is given by the series (Krall Fink, 1948):y n(x)=sum… …   Wikipedia

  • Bessel filter — In electronics and signal processing, a Bessel filter is a variety of linear filter with a maximally flat group delay (linear phase response). Bessel filters are often used in audio crossover systems. Analog Bessel filters are characterized by… …   Wikipedia