Bessel Functions

Bessel Functions


cylinder functions of the first kind; they occur in the consideration of physical processes (heat conduction, diffusion, oscillations, and so on) in areas with circular and cylindrical symmetry. They are solutions of the Bessel equation.

The Bessel function Jp of the order (index) p,- ∞ < p < ∞, is represented by the series

which converges for all x. Its graph for x > 0 has the form of a damped oscillation; Jp(x) has an infinite number of zeros. The behavior of Jp (x) for small | x | is given by the first term of the series (1); for large x > 0, the following asymptotic representation holds:

Here the oscillatory character of the function is clearly shown. The Bessel functions of the “half-integer” order p = n + ½ are expressed in terms of elementary functions—in particular,

The Bessel functions Bessel Functions, where Bessel Functions are the positive zeros of Jp(x) and where p > - ½, form an orthogonal system with weight × in the interval (0, l).

The function J0 was first considered by D. Bernoulli in a paper devoted to the oscillation of heavy chains (1732). L. Euler, considering the problem of the oscillations of a circular membrane (1738), arrived at the Bessel equation with whole values p = n and found an expression for Jn(x) in the form of a series of the powers of x. In later works he extended this expression to the case of arbitrary ρ values. F. Bessel investigated the functions Jp (x) in connection with the study of the motion of the planets around the sun (1824). He compiled the first tables for J0 (x), J1 (x), and J2 (x).


Watson, G. N. Teoriia besselevykh funktsii, parts 1–2. Moscow, 1949. (Translated from English.)
Lebedev, N. N. Spetsial’nye funktsii i ikh prilozheniia, 2nd ed. Moscow-Leningrad, 1963.
Bateman, H., and A. Erdély. Vysshie transtsendentnye funktsii: Funktsii Besselia, funktsii parabolicheskogo tsilindra, orto-gonal’nye mnogochleny. Moscow, 1966. (Translated from English.) P. I. Lizorkin
References in periodicals archive ?
It was Voronoi [13] who introduced a new phase not only into the lattice point problem but also into the fields where there is a zeta-function, as expressing the error term in terms of special functions, and in particular Bessel functions.
The author covers Bessel and associated functions, generating functions of Bessel and associated Bessel functions, convergence of series in Bessel functions, and a wide variety of related subjects.
m](r), m = 0,1,2--modified Bessel functions of the third kind (or Macdonald functions);
Laplaceequation, in terms of cylindrical Bessel functions Im(x) and Km(x), with m=0, [+ or -] 1, [+ or -] 2, [+ or -] 3,.
Janowski Starlikeness of Generalized Bessel Functions
Motivated by results on connection between various subclasses of analytic functions by using the hypergeometric function by many author particularly the authors (see[5]- [10]) and generalized Bessel functions (see [12]- [13]), S.
This Bessel function of the first kind and of order zero is used to produce Bessel functions of the first kind and orders 1, 2, 3, .
Appendices include physical properties of relevant materials and Bessel functions.
Abstract: Using Bessel functions of first kind we introduce new integral operators and show that these operators are in the class N(th).
7] RatisYuL, and Fernandez de Cordoba P, A code to evaluate (high order) Bessel functions based on the continued fractions method, Computer Physics Communications 1993; 76, 381-388.
4](k[DELTA]) and that they do not contain any odd term Bessel functions.
Then, from [6] and using some Bessel functions properties, we can easily find: