BESSEL FUNCTION

Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation:

for an arbitrary real number α (the order). The most common and important special case is where α is an integer n.

Although α and -α produce the same differential equation, it is conventional to define different Bessel functions for these two orders (e.g. so that the Bessel functions are analytic functions of α).

Applications

Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates, and Bessel functions are therefore especially important for many problems of wave propagation, static potentials, and so on. (For cylindrical problems, one obtains Bessel functions of integer order α = n; for spherical problems, one obtains half integer orders α = n+1/2.) For example:

electromagnetic waves in a cylindrical waveguide
heat conduction in a cylindrical object.
modes of vibration of a thin circular (or annular) membrane.

Bessel functions also have useful properties for other problems, such as signal processing (e.g. see FM synthesis or Kaiser window).

Definitions

Since this is a second-order differential equation, there must be two linearly independent solutions. Depending upon the circumstances, however, various formulations of these solutions are convenient, and the different variations are described below.

Bessel functions of the first and second kind

These are perhaps the most commonly used forms of the Bessel functions.

Bessel functions of the first kind, J_α(x), are solutions of Bessel's differential equation which are finite at x = 0 for α an integer or α non-negative. (The specific choice and normalization of J_α are defined by its properties below.)
Bessel functions of the second kind, Y_α(x), are solutions which are singular (infinite) at x = 0.

Y_α(x) is sometimes also called the Neumann function, and is occasionally denoted instead by N_α(x). It is related to J_α(x) by:

where the case of integer α is handled by taking the limit.

For integer order n, J_n and J_-n are not linearly independent:

J_-n(x) = (-1)ⁿ J_n(x),

Y_-n(x) = (-1)ⁿ Y_n(x),

in which case Y_n is needed to provide the second linearly independent solution of Bessel's equation. In contrast, for non-integer order, J_α and J_-α are linearly independent, and Y_α is redundant (as is clear from its definition above).

The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportional to 1/√x (see also their asymptotic forms, below), although their roots are not generally periodic except asymptotically for large x.

Plot of three Bessel functions of the first kind: J₀, J₁, and J₂.

Plot of three Bessel functions of the second kind: Y₀, Y₁, and Y₂.

Asymptotic Forms

The Bessel functions have the following asymptotic forms. For small arguments 0 < x << 1, one obtains:

where α is non-negative and Γ denotes the Gamma function. For large arguments x >> 1, they become:

Asymptotic forms for the other types of Bessel function follow straightforwardly from the above relations. For example, for large x >> 1, the modified Bessel functions become:

Properties

For integer order α = n, J_n is often defined via a Laurent series for a generating function:

an approach used by P. A. Hansen in 1843. (This can be generalized to non-integer order by contour integration or other methods.) Another important relation for integer orders is the Jacobi-Anger identity:

which is used to expand a plane wave as a sum of cylindrical waves.

The functions J_α, Y_α, H_α⁽¹⁾, and H_α⁽²⁾ all satisfy the recurrence relations:

Z_α-1(x) + Z_α+1(x) = 2α/x Z_α(x)

Z_α-1(x) - Z_α+1(x) = 2 dZ_α/dx

where Z denotes J, Y, H⁽¹⁾, or H⁽²⁾. (These two identities are often combined to yield various other relations.)

Because Bessel's equation becomes Hermitian (self-adjoint) if it is divided by x, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that:

where α > -1, δ_m,n is the Kronecker delta, and u_α,m is the m-th zero of J_α(x). This orthogonality relation can then be used to extract the coefficients in the Fourier-Bessel series, where a function is expanded in the basis of the functions J_α(x u_α,m) for fixed α and varying m. (An analogous relationship for the spherical Bessel functions follows immediately.)

Another orthogonality relation is the closure equation:

for α > -1/2 and where δ is the Dirac delta function.

Another important consequence of the Hermitian nature of Bessel's equations involves the Wronskian of the solutions:

where A_α and B_α are any two solutions of Bessel's equation, and C_α is a constant independent of x (which depends on α and on the particular Bessel functions considered). For example, if A_α = J_α and B_α = Y_α, then C_α is 2/&pi. This also holds for the modified Bessel functions; for example, if A_α = I_α and B_α = K_α, then C_α is -1.

(There are a large number of other known integrals and identities that are not reproduced here, but which can be found in the references.)

References

George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists (Harcourt: San Diego, 2001).
Frank Bowman, Introduction to Bessel Functions (Dover: New York, 1958).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover: New York, 1972).

Scrabble® Enable2K-Verified Anagrams
	Words within the letters "b-c-e-e-f-i-l-n-n-o-s-s-t-u"
	-2 letters: functionless.
	-4 letters: bluestones, confluents, flounciest, futileness, influences, insolences, nonbeliefs, ostensible, selections, stoneflies, sublicense, subsection.
	-5 letters: blousiest, bluenoses, bluesiest, bluestone, bluntness, boletuses, bounciest, buntlines, bustlines, cenobites, confluent, continues, countless, elections, esculents, felstones, fencibles, flections, fluencies, functions, influence, influents, insolence, insolents, loftiness, nebulises, nescients, nobleness, nonbelief, obscenest, seclusion, seicentos, selection, sentinels, stilbenes, subtonics, tubenoses, unbeliefs.
Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.

Domain	Definition
Electrical Engineering	A mathematical function used to yield a maximally constant time delay in a filter, with little if any consideration for amplitude response. Source: European Union. (references)
Source: compiled by the editor from various references; see credits.

Expression	Frequency per Day
bessel function	23
bessel function modified	4
bessel function rational	2
bessel function kapteyn series	2
bessel function table	2
Source: compiled by the editor from various references; see credits.

Language		Translations for "BESSEL FUNCTION"; alternative meanings/domain in parentheses.
Hungarian		bessel-függvény. (various references)

Pig Latin		esselbay unctionfay
Source: compiled by the editor from various translation references.

BESSEL FUNCTION

Specialty Definition: BESSEL FUNCTION

Electrical Engineering

Specialty Definition: Bessel function

Applications

Definitions

Bessel functions of the first and second kind

Hankel functions

Modified Bessel functions

Spherical Bessel functions

Asymptotic Forms

Properties

References

Synonym: BESSEL FUNCTION

Expressions: BESSEL FUNCTION

Frequency of Internet Keywords: BESSEL FUNCTION

bessel function

bessel function modified

bessel function rational

bessel function kapteyn series

bessel function table

Modern Translation: BESSEL FUNCTION

Hungarian

Pig Latin

Anagrams: BESSEL FUNCTION

Scrabble® Enable2K-Verified Anagrams

Alternative Orthography: BESSEL FUNCTION

42 45 53 53 45 4C 46 55 4E 43 54 49 4F 4E

01000010 01000101 01010011 01010011 01000101 01001100 00100000 01000110 01010101 01001110 01000011 01010100 01001001 01001111 01001110

&#66 &#69 &#83 &#83 &#69 &#76 &#32 &#70 &#85 &#78 &#67 &#84 &#73 &#79 &#78

0042 0045 0053 0053 0045 004C 0046 0055 004E 0043 0054 0049 004F 004E

36395353394624055483754434948

INDEX

Expressions using "BESSEL FUNCTION": Bessel function distribution ♦ Bessel function of zero order. Additional references.
Source: compiled by the editor from various references; see credits.