Bessel function

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions $y (x)$ of Bessel's differential equation

The most important cases are when $\alpha$ is an integer or half-integer. Bessel functions for integer $\alpha$ are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer $\alpha$ are obtained when solving the Helmholtz equation in spherical coordinates.

in a cylindrical waveguide

Electromagnetic waves

Pressure amplitudes of rotational flows

inviscid

in a cylindrical object

Heat conduction

Modes of vibration of a thin circular or annular (such as a drumhead or other membranophone) or thicker plates such as sheet metal (see Kirchhoff–Love plate theory, Mindlin–Reissner plate theory)

acoustic membrane

Diffusion problems on a lattice

Solutions to the radial (in spherical and cylindrical coordinates) for a free particle

Schrödinger equation

Position space representation of the Feynman in quantum field theory

propagator

Solving for patterns of acoustical radiation

Frequency-dependent friction in circular pipelines

Dynamics of floating bodies

Angular resolution

Diffraction from helical objects, including

DNA

of product of two normally distributed random variables^[1]

Probability density function

Analyzing of the surface waves generated by microtremors, in and seismology.

geophysics

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 ( $α = n$ ); in spherical problems, one obtains half-integer orders ( $α = n + 1 / 2$ ). For example:

Bessel functions also appear in other problems, such as signal processing (e.g., see FM audio synthesis, Kaiser window, or Bessel filter).

Basset function after

Alfred Barnard Basset

Modified Bessel function of the third kind

Modified Hankel function

[29]

Macdonald function after

Hector Munro Macdonald

Zeros of the Bessel function[edit]

Bourget's hypothesis[edit]

Bessel himself originally proved that for nonnegative integers $n$ , the equation $J n (x) = 0$ has an infinite number of solutions in $x$ .^[55] When the functions $J n (x)$ are plotted on the same graph, though, none of the zeros seem to coincide for different values of $n$ except for the zero at $x = 0$ . This phenomenon is known as Bourget's hypothesis after the 19th-century French mathematician who studied Bessel functions. Specifically it states that for any integers $n \geq 0$ and $m \geq 1$ , the functions $J n (x)$ and $J n + m (x)$ have no common zeros other than the one at $x = 0$ . The hypothesis was proved by Carl Ludwig Siegel in 1929.^[56]

Transcendence[edit]

Siegel proved in 1929 that when ν is rational, all nonzero roots of $J ν (x)$ and $J' ν (x)$ are transcendental,^[57] as are all the roots of $K ν (x)$ .^[52] It is also known that all roots of the higher derivatives $J_{\nu }^{(n)}(x)$ for $n \leq 18$ are transcendental, except for the special values $J_{1}^{(3)}(\pm {\sqrt {3}})=0$ and $J_{0}^{(4)}(\pm {\sqrt {3}})=0$ .^[57]

Numerical approaches[edit]

For numerical studies about the zeros of the Bessel function, see Gil, Segura & Temme (2007), Kravanja et al. (1998) and Moler (2004).

Numerical values[edit]

The first zero in J₀ (i.e, j_0,1, j_0,2 and j_0,3) occurs at arguments of approximately 2.40483, 5.52008 and 8.65373, respectively.^[58]