[permalink] [id link]
The Bessel function of the second kind then can be thought to naturally appear as the imaginary part of the Hankel functions.
from
Wikipedia
Some Related Sentences
Bessel and function
where is the cylindrical Bessel function of the first kind and are undetermined constants.
for an arbitrary real or complex number α ( the order of the Bessel function ); the most common and important cases are for α an integer or half-integer.
Plot of Bessel function of the first kind, J < sub > α </ sub >( x ), for integer orders α = 0, 1, 2.
In this case, 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, for integer values of, is possible using an integral representation:
This was the approach that Bessel used, and from this definition he derived several properties of the function.
This expression is related to the development of Bessel functions in terms of the Bessel – Clifford function.
In terms of the Laguerre polynomials and arbitrarily chosen parameter, the Bessel function can be expressed as
Plot of Bessel function of the second kind, Y < sub > α </ sub >( x ), for integer orders α = 0, 1, 2.
This orthogonality relation can then be used to extract the coefficients in the Fourier – Bessel series, where a function is expanded in the basis of the functions J < sub > α </ sub >( x u < sub > α, m </ sub >) for fixed α and varying m.
This property is used to construct an arbitrary function from a series of Bessel functions by means of the Hankel transform.
* Bessel – Clifford function
* Wright generalized Bessel function
* Wolfram function pages on Bessel J and Y functions, and modified Bessel I and K functions.
where a is the radius of the circular aperture, k is equal to 2π / λ and J < sub > 1 </ sub > is a Bessel function.
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.
For more general distributions the asymptotic distribution can be expressed as a Bessel function.
For a circular aperture, the diffraction-limited image spot is known as an Airy disk ; the distance x in the single-slit diffraction formula is replaced by radial distance r and the sine is replaced by 2J < sub > 1 </ sub >, where J < sub > 1 </ sub > is a first order Bessel function.
If B is a circle, then these eigenfunctions have an angular component that is a trigonometric function of the polar angle θ, multiplied by a Bessel function ( of integer order ) of the radial component.
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 ):
The red lines in these three graphs correspond to the simulated images above, and the green lines were computed by applying the corresponding parameters to the squared Bessel function given above.
Bessel and second
The Bessel functions of the second kind, denoted by Y < sub > α </ sub >( x ), occasionally denoted instead by N < sub > α </ sub >( x ), are solutions of the Bessel differential equation.
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.
Bessel and kind
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.
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
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.
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,.
0.157 seconds.