[permalink] [id link]
As Bessel, Legendre and Gauss did, Kramp worked on the generalised factorial function which applied to non-integers.
from
Wikipedia
Some Related Sentences
Bessel and Legendre
* Lattice AN6017 application note comparing and contrasting filters ( in order of damping coefficient, from lower to higher values ): Gaussian, Bessel, linear phase, Butterworth, Chebyshev, Legendre, elliptic.
It was introduced by Legendre and Bessel who solved problems for geodesics on the ellipsoid by transforming them to an equivalent problem for spherical geodesics by using this smaller latitude.
He gave expressions for the Bessel functions as integrals involving Legendre functions.
Bessel and Gauss
Further proofs were given by Laplace ( 1810, 1812 ), Gauss ( 1823 ), James Ivory ( 1825, 1826 ), Hagen ( 1837 ), Friedrich Bessel ( 1838 ), W. F. Donkin ( 1844, 1856 ), and Morgan Crofton ( 1870 ).
Encke sent his calculations as a note to Gauss, Olbers, and Bessel.
Gauss, Stirling, and Bessel all developed formulae to remedy that problem.
The other difference formulas, such as those of Stirling, Bessel and Gauss, can be derived from Newton's, using Newton's terms, with data points and x values renamed in keeping with the choice of x zero, and based on the fact that they must add up to the same sum value as Newton's ( With Stirling that is so when polynomial degree is even.
Examples of complete exponential sums are Gauss sums and Kloosterman sums ; these are in some sense finite field or finite ring analogues of the gamma function and some sort of Bessel function, respectively, and have many ' structural ' properties.
After the 1772 and 1805 apparitions a number of attempts were made by Lalande ( 1774 ), Gauss ( 1806 ) and Bessel ( 1806 ) to calculate a definitive orbit, Gauss and Olbers both noting a similarity between the 1805 and 1772 comets, but it was not possible to prove a link.
In his report, Kater mentioned previous work in this area by Carl Friedrich Gauss and Friedrich Bessel.
Bessel and did
Schumacher encouraged him to study astronomy and Peters did a PhD under Friedrich Bessel at the University of Königsberg.
The investigation under Robeson, however, did not charge Dr. Bessel with Hall ’ s murder, although there was circumstantial evidence Dr. Bessel did murder Hall ; however, the entire Polaris expedition crew was exonerated.
This was the 6. 2 " ( 157 mm ) aperture Fraunhofer heliometer at Königsberg Observatory built by Joseph von Fraunhofer's firm, though he did not live to see it delivered to Bessel.
Bessel and worked
The Board of Trustees resolved that it would be named for his father, Franz Theodore Stone, a Prussian mathematician and astronomical researcher who worked for Friedrich Wilhelm Bessel after attending the Königsberg Albertina University.
The filters are also called Bessel – Thomson filters in recognition of W. E. Thomson, who worked out how to apply Bessel functions to filter design.
Many important astronomers, such as Friedrich Wilhelm Bessel, Friedrich Wilhelm Argelander and Arthur Auwers worked there.
Bessel and on
The equation on the left is the Bessel equation which has the general solution
where A < sub > α </ sub > and B < sub > α </ sub > are any two solutions of Bessel's equation, and C < sub > α </ sub > is a constant independent of x ( which depends on α and on the particular Bessel functions considered ).
* Watson, G. N., A Treatise on the Theory of Bessel Functions, Second Edition, ( 1995 ) Cambridge University Press.
* Wolfram function pages on Bessel J and Y functions, and modified Bessel I and K functions.
Bessel came to the attention of a major figure of German astronomy at the time, Heinrich Wilhelm Olbers, by producing a refinement on the orbital calculations for Halley's Comet.
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.
In addition, Bessel showed that if the pendulum was made with a symmetrical shape, but internally weighted on one end, the error caused by effects of air resistance would cancel out.
Second-order filters can have a Bessel, Linkwitz-Riley or Butterworth characteristic depending on design choices and the components used.
Mostly on the recommendation of Bessel, Encke became director of the new observatory and secretary of the Academy of Sciences.
* D. McGloin on Review of Bessel beam optical tweezers
This article relies heavily on Bessel functions and Laguerre polynomials.
In his section on mathematical physics Neumann taught mathematical methods and as well as the techniques of an exact experimental physics grounded in the type of precision measurement perfected by his astronomer colleague Friedrich Wilhelm Bessel.
By convention these features are identified on lunar maps by placing the letter on the side of the crater midpoint that is closest to Bessel.
His Treatise on the theory of Bessel functions ( 1922 ) was a virtuoso display, in particular in the asymptotic expansions of Bessel functions.
Early researches by Friedrich Wilhelm Bessel ( 1784 – 1846 ) in Königsberg and Adolf Hirsch led to the development of a highly precise chronoscope by Mathias Hipp that, in turn, was based on a design by Charles Wheatstone for a device that measured the speed of artillery shells ( Edgell & Symes, 1906 ).
The map projection used is Oblique Mercator on an 1841 Bessel ellipsoid.
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.
The Bessel function of the second kind then can be thought to naturally appear as the imaginary part of the Hankel functions.
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
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 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.
0.285 seconds.