Bessel function

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

$x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+(x^{2}-\alpha ^{2})y=0$

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

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. (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
• modes of vibration of a thin circular (or annular) membrane.
• the separability of an optical instruments.

See also: FM synthesis

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 normalization of J_α is 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:

$Y_{\alpha }(x)={\frac {J_{\alpha }(x)\cos(\alpha \pi )-J_{-\alpha }(x)}{\sin(\alpha \pi )}},$

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),

in which case Y_n is required to get the second linearly independent solution of Bessel's equation.

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 periodic except asymptotically for large x.

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:

H_α⁽¹⁾(x) = J_α(x) + i Y_α(x)

H_α⁽²⁾(x) = J_α(x) - i Y_α(x)

where i is the imaginary unit. (The Hankel functions express inward- and outward-propagating cylindrical wave solutions of the cylindrical wave equation.)

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 of the first and second kind, and are defined by:

I_α(x) = i^-α J_α(ix)

K_α(x) = π i^α+1 H_α⁽¹⁾(ix) / 2

These are chosen to be real-valued for real arguments x. Unlike the ordinary Bessel functions, which are oscillating, I_α and K_α are exponentially growing and decaying functions, respectively. They are the two linearly independent solutions to the modified Bessel's equation:

$x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}-(x^{2}+\alpha ^{2})y=0.$

Spherical Bessel functions

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

$x^{2}{\frac {d^{2}y}{dx^{2}}}+2x{\frac {dy}{dx}}+(x^{2}-n(n+1))y=0.$

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_α and Y_α by:

$j_{n}(x)={\sqrt {\frac {\pi }{2x}}}J_{n+1/2}(x),$

$y_{n}(x)={\sqrt {\frac {\pi }{2x}}}Y_{n+1/2}(x)=(-1)^{n+1}{\sqrt {\frac {\pi }{2x}}}J_{-n-1/2}(x).$

There are also spherical analogues of the Hankel functions:

h_n⁽¹⁾(x) = j_n(x) + i y_n(x)

h_n⁽²⁾(x) = j_n(x) - i y_n(x)

Asymptotic Forms

The Bessel functions have the following asymptotic forms:

$J_{\alpha }(x)\rightarrow {\frac {1}{\Gamma (\alpha +1)}}\left({\frac {x}{2}}\right)^{\alpha }$

$Y_{\alpha }(x)\rightarrow \left\{{\begin{matrix}{\frac {2}{\pi }}\ln(x/2)&{\mbox{if }}\alpha =0\\\\-{\frac {\Gamma (\alpha )}{\pi }}\left({\frac {2}{x}}\right)^{\alpha }&{\mbox{if }}\alpha >0\end{matrix}}\right.$

for 0 < x << 1, non-negative α, and where Γ denotes the Gamma function. For large arguments x >> 1, they become:

$J_{\alpha }(x)\rightarrow {\sqrt {\frac {2}{\pi x}}}\cos \left(x-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)$

$Y_{\alpha }(x)\rightarrow {\sqrt {\frac {2}{\pi x}}}\sin \left(x-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right).$

Properties

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) = 2&alpha/x Z_α(x)

Z_α-1(x) - Z_α+1(x) = 2 dZ_α/dx

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References:

• George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists (Harcourt: San Diego, 2001).