Bessel function


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 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:
Bessel functions also appear in other problems, such as signal processing.

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.
TypeFirst kindSecond 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, which can be found by applying the Frobenius method to Bessel's equation:
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 ; 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 , although their roots are not generally periodic, except asymptotically for large.
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 :
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, 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 :

Relation to hypergeometric series

The Bessel functions can be expressed in terms of the generalized hypergeometric series as
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 series
where is the digamma function, the logarithmic derivative of the gamma function.
There is also a corresponding integral formula :
In the case where :
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 is 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 as
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.
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:
In particular, if with a nonnegative integer, the above relations imply directly that
These are useful in developing the spherical Bessel functions.
The Hankel functions admit the following integral representations for :
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 'of the first and second kind and are defined as
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 :
and are the two linearly independent solutions to the modified Bessel's equation:
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.
Two integral formulas for the modified Bessel functions are :
Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example :
It can be proven by showing equality to the above integral definition for. This is done by integrating a closed curve in the first quadrant of the complex plane.
Modified Bessel functions of the second kind may be represented with Bassett's integral
Modified Bessel functions and can be represented in terms of rapidly convergent integrals
The modified Bessel function is useful to represent the Laplace distribution as an Exponential-scale mixture of normal distributions.
The modified Bessel function of the second kind has also been called by the following names :