Bessel function

Bessel functions, invented by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are 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.

Bessel functions of the first and second kind

The solutions come in two linearly independent kinds, with other variations described below.

Bessel functions of the first kind, J_α(x), are solutions of the above differential equation which are finite at x = 0.
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.

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

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.

Asymptotic Forms

The ordinary Bessel functions have the following asymptotic forms for large arguments:

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)

for x >> 1.

(Asymptotic forms for large and small arguments are known for all the types of Bessel function.)

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 example:

electromagnetic waves in a cylindrical waveguide
modes of vibration of a thin circular (or annular) membrane.
the separability of an optical instruments.