Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena with circular or cylindrical symmetry. They are named after the German astronomer and mathematician Friedrich Bessel, who studied them systematically in 1824.
Bessel functions are solutions to a particular type of ordinary differential equation: where is a number that determines the shape of the solution. This number is called the order of the Bessel function and can be any complex number. Although the same equation arises for both and , mathematicians define separate Bessel functions for each to ensure the functions behave smoothly as the order changes.
The most important cases are when is an integer or a half-integer. When is an integer, the resulting Bessel functions are often called cylinder functions or cylindrical harmonics because they naturally arise when solving problems (like Laplace's equation) in cylindrical coordinates. When is a half-integer, the solutions are called spherical Bessel functions and are used in spherical systems, such as in solving the Helmholtz equation in spherical coordinates.
Applications Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are therefore especially important for many problems of wave propagation and static potentials. In solving problems in cylindrical coordinate systems, one obtains Bessel functions of integer order (); in spherical problems, one obtains half-integer orders (). For example: Electromagnetic waves in a cylindrical waveguide Pressure amplitudes of inviscid rotational flows Heat conduction in a cylindrical object Modes of vibration of a thin circular or annular acoustic membrane (such as a drumhead or other membranophone) or thicker plates such as sheet metal (see Kirchhoff–Love plate theory, Mindlin–Reissner plate theory) Diffusion problems on a lattice Solutions to the Schrödinger equation in spherical and cylindrical coordinates for a free particle Position space representation of the Feynman propagator in quantum field theory Solving for patterns of acoustical radiation Frequency-dependent friction in circular pipelines Dynamics of floating bodies Angular resolution X-ray crystallography of helical structures, including DNA and the alpha helix of proteins Probability density function of product of two normally distributed random variables Analyzing of the surface waves generated by microtremors, in geophysics and seismology.
Bessel functions also appear in other fields, such as signal processing (e.g., see FM audio synthesis, Kaiser window, or Bessel filter). They also appear in pure mathematics as part of the Fourier expansion of Maass forms.
Definitions Because this is a linear differential equation, solutions can be scaled to any amplitude. The amplitudes chosen for the functions originate from the early work in which the functions appeared as solutions to definite integrals rather than solutions to differential equations. Because the differential equation is second-order, there must be two linearly independent solutions: one of the first kind and one of the second kind. Depending upon the circumstances, however, various formulations of these solutions are convenient. Different variations are summarized in the table below and described in the following sections. The subscript n is typically used in place of when is known to be an integer.
Type First kind Second kind Bessel functions Modified Bessel functions Hankel functions Spherical Bessel functions Modified spherical Bessel functions Spherical Hankel functions
Bessel functions of the second kind and the spherical Bessel functions of the second kind are sometimes denoted by and , respectively, rather than and .
Bessel functions of the first kind: Jα
Bessel functions of the first kind, denoted as , are solutions of Bessel's differential equation. For integer or positive , Bessel functions of the first kind are finite at the origin (); while for negative non-integer , Bessel functions of the first kind diverge as approaches zero. It is possible to define the function by times a Maclaurin series (note that need not be an integer, and non-integer powers are not permitted in a Taylor series), which can be found by applying the Frobenius method to Bessel's equation:Abramowitz and Stegun, p. 360, 9.1.10.
where is the gamma function, a shifted generalization of the factorial function to non-integer values. Some earlier authors define the Bessel function of the first kind differently, essentially without the division by in ; For example, Hansen (1843) and Schlömilch (1857). this definition is not used in this article. The Bessel function of the first kind is an entire function if is an integer, otherwise it is a multivalued function with singularity at zero. The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportionally to (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large . (The series indicates that is the derivative of , much like is the derivative of ; more generally, the derivative of can be expressed in terms of by the identities below.)
For non-integer , the functions and are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order , the following relationship is valid (the gamma function has simple poles at each of the non-positive integers):Abramowitz and Stegun, p. 358, 9.1.5.
This means that the two solutions are no longer linearly independent. In this case, the second linearly independent solution is then found to be the Bessel function of the second kind, as discussed below.
Bessel's integrals Another definition of the Bessel function, for integer values of , is possible using an integral representation:
which is also called Hansen-Bessel formula.
This was the approach that Bessel used,Bessel, F. (1824). The relevant integral is an unnumbered equation between equations 28 and 29. Note that Bessel's would today be written . and from this definition he derived several properties of the function. The definition may be extended to non-integer orders by one of Schläfli's integrals, for :Watson, p. 176Arfken & Weber, exercise 11.1.17.
Relation to hypergeometric series The Bessel functions can be expressed in terms of the generalized hypergeometric series asAbramowitz and Stegun, p. 362, 9.1.69.
This expression is related to the development of Bessel functions in terms of the Bessel–Clifford function.
Relation to Laguerre polynomials In terms of the Laguerre polynomials and arbitrarily chosen parameter , the Bessel function can be expressed as
Bessel functions of the second kind: Yα
The Bessel functions of the second kind, denoted by , occasionally denoted instead by , are solutions of the Bessel differential equation that have a singularity at the origin () and are multivalued. These are sometimes called Weber functions, as they were introduced by , and also Neumann functions after Carl Neumann.
For non-integer , is related to by
In the case of integer order , the function is defined by taking the limit as a non-integer tends to :
If is a nonnegative integer, we have the seriesNIST Digital Library of Mathematical Functions, (10.8.1). Accessed on line Oct. 25, 2016.
where is the digamma function, the logarithmic derivative of the gamma function.
There is also a corresponding integral formula (for ):Watson, p. 178.
In the case where : (with being Euler's constant)
is necessary as the second linearly independent solution of the Bessel's equation when is an integer. But has more meaning than that. It can be considered as a "natural" partner of . See also the subsection on Hankel functions below.
When is an integer, moreover, as was similarly the case for the functions of the first kind, the following relationship is valid:
Both and are holomorphic functions of on the complex plane cut along the negative real axis. When is an integer, the Bessel functions are entire functions of . If is held fixed at a non-zero value, then the Bessel functions are entire functions of .
The Bessel functions of the second kind, when is an integer, are an example of the second kind of solution in Fuchs's theorem.
Hankel functions: H, H
Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions of the first and second kind, and , defined asAbramowitz and Stegun, p. 358, 9.1.3, 9.1.4.
where is the imaginary unit. These linear combinations are also known as Bessel functions of the third kind; they are two linearly independent solutions of Bessel's differential equation. They are named after Hermann Hankel.
These forms of linear combination satisfy numerous simple-looking properties, like asymptotic formulae or integral representations. Here, "simple" means an appearance of a factor of the form . For real where , are real-valued, the Bessel functions of the first and second kind are the real and imaginary parts, respectively, of the first Hankel function and the real and negative imaginary parts of the second Hankel function. Thus, the above formulae are analogs of Euler's formula, substituting , for and , for , , as explicitly shown in the asymptotic expansion.
The Hankel functions are used to express outward- and inward-propagating cylindrical-wave solutions of the cylindrical wave equation, respectively (or vice versa, depending on the sign convention for the frequency).
Using the previous relationships, they can be expressed as
If is an integer, the limit has to be calculated. The following relationships are valid, whether is an integer or not:Abramowitz and Stegun, p. 358, 9.1.6.
In particular, if with a nonnegative integer, the above relations imply directly that
These are useful in developing the spherical Bessel functions (see below).
The Hankel functions admit the following integral representations for :Abramowitz and Stegun, p. 360, 9.1.25.
where the integration limits indicate integration along a contour that can be chosen as follows: from to 0 along the negative real axis, from 0 to along the imaginary axis, and from to along a contour parallel to the real axis.
Modified Bessel functions: Iα, Kα The Bessel functions are valid even for complex arguments , and an important special case is that of a purely imaginary argument. In this case, the solutions to the Bessel equation are called the modified Bessel functions (or occasionally the hyperbolic Bessel functions) of the first and second kind and are defined asAbramowitz and Stegun, p. 375, 9.6.2, 9.6.10, 9.6.11.
when is not an integer. When is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments . The series expansion for is thus similar to that for , but without the alternating factor.
can be expressed in terms of Hankel functions:
Using these two formulae the result to commonly known as Nicholson's integral or Nicholson's formula, can be obtained to give the following
given that the condition is met. It can also be shown that
only when and but not when .
We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if ):Abramowitz and Stegun, p. 375, 9.6.3, 9.6.5.
and are the two linearly independent solutions to the modified Bessel's equation:Abramowitz and Stegun, p. 374, 9.6.1.
Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, and are exponentially growing and decaying functions respectively. Like the ordinary Bessel function , the function goes to zero at for and is finite at for . Analogously, diverges at with the singularity being of logarithmic type for , and otherwise.