Bessel function: Encyclopedia II - Bessel function - Definitions

Bessel function - 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 function - Bessel functions of the first kind

Bessel functions of the first kind, denoted with 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; another possibility is to define it by its Taylor series expansion around x = 0 (or a more general power series for non-integer α):

Here, Γ(z) is the gamma function, a generalization of the factorial to non-integer values. The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportionally to 1/√x (see also their asymptotic forms, below), although their roots are not generally periodic except asymptotically for large x.

Here is the plot of J_α(x) for α = 0,1,2:

If α is not an integer, the functions J_α(x) and J _{− α}(x) are linearly independent and are therefore the two solutions of the differential equation. On the other hand, if the order α is an integer, then the following relationship is valid:

This means that they are no longer linearly independent. The second linearly independent solution is then found to be the Bessel function of the second kind, as discussed below.

Another definition of the Bessel function is possible using an integral equation:

This is the approach that Bessel used, and from this definition he derived several properties of the function. Another integral representation is:

The Bessel functions can be expressed in terms of the hypergeometric series as

Bessel function - Bessel functions of the second kind

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

The Bessel functions of the second kind, denoted by Y_α(x), are solutions of the Bessel differential equation. They 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.

When α is not an integer, the definition of Y_α is redundant (as is clear from its definition above). On the other hand, when α is an integer, Y_α is the second linearly independent solution of Bessel's equation; moreover, as was similarly the case for the functions of the first kind, the following relationship is valid:

Both J_α(x) and Y_α(x) are holomorphic functions of x on the complex plane cut along the negative real axis. When α is an integer, there is no branch point, and the Bessel functions are entire functions of x. If x is held fixed, then the Bessel functions are entire functions of α.

Here is the plot of Y_α(x) for α = 0,1,2:

Bessel function - Hankel functions

Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions H_α⁽¹⁾(x) and H_α⁽²⁾(x), defined by:

where i 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. The Hankel functions express inward- and outward-propagating cylindrical wave solutions of the cylindrical wave equation. They are named for Hermann Hankel.

Using the previous relationships they can be expressed as:

if α is an integer, the limit has to be calculated. The following relationships are valid, whether α is an integer or not:

Bessel function - Modified Bessel functions

The Bessel functions are valid even for complex arguments x, 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 by:

These are chosen to be real-valued for real arguments x. They are the two linearly independent solutions to the modified Bessel's equation:

Unlike the ordinary Bessel functions, which are oscillating, I_α and K_α are exponentially growing and decaying functions, respectively. Like the ordinary Bessel function J_α, the function I_α goes to zero at x=0 for α > 0 and is finite at x=0 for α=0. Analogously, K_α diverges at x=0.

The modified Bessel function of the second kind has also been called by the now-rare names:

• Basset function
• modified Bessel function of the third kind
• MacDonald function

Bessel function - Spherical Bessel functions

When solving for separable solutions of Laplace's equation in spherical coordinates, the radial equation has the form:

The two linearly independent solutions to this equation are called the spherical Bessel functions j_n and y_n (also denoted n_n), and are related to the ordinary Bessel functions J_n and Y_n by:

The spherical Bessel functions can also be written as:

The first spherical Bessel function j₀(x) is also known as the sinc function. The first few spherical Bessel functions are:

and

There are also spherical analogues of the Hankel functions:

In fact, there are simple closed-form expressions for the Bessel functions of half-integer order in terms of the standard trigonometric functions, and therefore for the spherical Bessel functions. In particular, for non-negative integers n:

and h_n⁽²⁾ is the complex-conjugate of this (for real x). (!! is the double factorial.) It follows, for example, that j₀(x) = sin(x)/x and y₀(x) = -cos(x)/x, and so on.

Bessel function - Riccati-Bessel functions

Riccati-Bessel functions only slightly differ from spherical Bessel functions:

They satisfy the differential equation:

This differential equation, and the Riccati-Bessel solutions, arises in the problem of scattering of electromagnetic waves by a sphere, known as Mie scattering after the first published solution by Mie (1908). See e.g. Du (2004) for recent developments and references.

Following Debye (1909), the notation ψ_n,χ_n is sometimes used instead of S_n,C_n.

Adapted from the Wikipedia article "Definitions", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki