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

Although α and − α produce the same differential equation, it is conventional to define different Bessel functions for these two orders ( e. g., so that the Bessel functions are mostly smooth functions of α ).

Bessel functions are also known as cylinder functions or cylindrical harmonics because they are found in the solution to Laplace's equation in cylindrical 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

Bessel functions also have useful properties for other problems, such as signal processing ( e. g., see FM synthesis, Kaiser window, or Bessel filter ).

Bessel functions of the first kind, denoted as J < sub > α </ sub >( x ), are solutions of Bessel's differential equation that are finite at the origin ( x = 0 ) for integer α, and diverge as x approaches zero for negative non-integer α.

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.

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.

When α is an integer, the Bessel functions J are entire functions of x.

If x is held fixed, then the Bessel functions are entire functions of α.

In this case, the second linearly independent solution is then found to be the Bessel function of the second kind, as discussed below.

Plot of Bessel function of the second kind, Y < sub > α </ sub >( x ), for integer orders α = 0, 1, 2.

The Bessel function of the second kind then can be thought to naturally appear as the imaginary part of the Hankel functions.

In the second decade of the 19th century while studying the dynamics of ' many-body ' gravitational systems, Bessel developed what are now known as Bessel functions.

Bessel didn't construct such a pendulum, but in 1864 Adolf Repsold, under contract to the Swiss Geodetic Commission, developed a symmetric pendulum 56 cm long with interchangeable pivot blades, with a period of about 3 / 4 second.

, where the function is a modified Bessel function of the second kind.

which links Bessel functions to ; this reduces to Kummer's second formula for:

( If a is a positive integer, the independent solution is given by the appropriate Bessel function of the second kind.

where K < sub > j </ sub > is the modified Bessel function of the second kind.

where g is the acceleration due to gravity, I is the second moment of area of the beam cross section, and B is the first zero of the Bessel function of the first kind of order-1 / 3, which is equal to 1. 86635 ...

# REDIRECT Bessel function # Bessel functions of the second kind: Yα

where is the modified Bessel function of the second kind.

where T is absolute temperature in kelvins, is the kinematic viscosity in centistokes, is the zero order modified Bessel function of the second kind, and A and B are liquid specific values.

where K < sub > p </ sub > is a modified Bessel function of the second kind, a > 0, b > 0 and p a real parameter.

where is the cylindrical Bessel function of the first kind and are undetermined constants.

Plot of Bessel function of the first kind, J < sub > α </ sub >( x ), for integer orders α = 0, 1, 2.

These linear combinations are also known as Bessel functions of the third kind ; they are two linearly independent solutions of Bessel's differential equation.

* Wolfram Mathworld – Bessel functions of the first kind

For the case of a carrier modulated by a single sine wave, the resulting frequency spectrum can be calculated using Bessel functions of the first kind, as a function of the sideband number and the modulation index.

In March 2011, Chinese scientists posited that a specific type of Bessel beam ( a special kind of laser that that does not diffract at the centre ) is capable of creating a pull-like effect on a given microscopic particle, forcing it towards the beam source.

The lateral intensity distribution on the screen has in fact the shape of a squared zeroth Bessel function of the first kind when close to the optical axis and using a plane wave source ( point source at infinity ):

where, is the pressure on axis, is the piston radius, is the wavelength ( i. e. ) is the angle off axis and is the Bessel function of the first kind.

The normalised autocorrelation function of a Rayleigh faded channel with motion at a constant velocity is a zeroth-order Bessel function of the first kind:

where is the 0th order modified Bessel function of the first kind.

* I < sub > 0 </ sub > is the zeroth order Modified Bessel function of the first kind.

which regular solutions for positive energies are given by so-called Bessel functions of the first kind ' so that the solutions written for R are the so-called Spherical Bessel function

The physical argument that ψ must be defined everywhere selected Bessel function of the first kind J over the other possibilities in the vacuum case.

where is the maximum intensity of the pattern at the Airy disc center, is the Bessel function of the first kind of order first, is the wavenumber, is the radius of the aperture, and is the angle of observation, i. e. the angle between the axis of the circular aperture and the line between aperture center and observation point.

* The unnormalized sinc is the zero < sup > th </ sup > order spherical Bessel function of the first kind,.