Infobox scientist name FriedrichBessel image Friedrich Wilhelm Bessel 1839 painting .jpg image size 225px caption Christian Albrecht Jensen C. A. Jensen , Friedrich Wilhelm Bessel , 1839 Ny Carlsberg ... of the Royal Astronomical Society 1829 & 1841 religion footnotes Friedrich Wilhelm Bessel 22 July ... last Fricke title Bessel, Friedrich Wilhelm volume 2 pages 97&ndash 102 References Reflist John ... on line de ADB 2 558 567 Bessel, Friedrich Wilhelm Karl Christian Bruhns ADB Bessel, Friedrich Wilhelm on line Cite NIE Bessel, Friedrich Wilhelm year 1905 External links MacTutor Biography id Bessel MathGenealogy id 18603 Cite Nuttall Bessel, Friedrich Wilhelm first letter B Persondata Metadata see Wikipedia Persondata . NAME Bessel, Friedrich ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH 22 ... OF DEATH K nigsberg , Province of Prussia Prussia , now Kaliningrad, Russia DEFAULTSORT Bessel, Friedrich ... Wilhelm Bessel ast FriedrichBessel bn ca FriedrichBessel cs Friedrich Wilhelm Bessel da Friedrich Wilhelm Bessel de Friedrich Wilhelm Bessel el es FriedrichBessel eo FriedrichBessel fa fr Friedrich Wilhelm Bessel gl FriedrichBessel ko id FriedrichBessel it Friedrich Wilhelm Bessel he kk ht FriedrichBessel lb Friedrich Wilhelm Bessel hu FriedrichBessel mk nl FriedrichBessel ja no Friedrich Wilhelm Bessel pl Friedrich Wilhelm Bessel pt Friedrich Wilhelm Bessel ro Friedrich Wilhelm Bessel ru , scn Friedrich Wilhelm Bessel sk Friedrich Wilhelm Bessel sl Friedrich Wilhelm Bessel sr fi FriedrichBessel sv Friedrich Wilhelm Bessel th tr Friedrich Wilhelm Bessel uk vi FriedrichBessel zh ... mater Georg August Universit t G ttingen Georg August University doctoral advisor Carl Friedrich Gauss ... more details
Bessel may refer to Bessel beam Bessel ellipsoid Bessel function in mathematics Bessel s inequality in mathematics Bessel filter , a linear filter often used in audio crossover systems Bessel crater , a small lunar crater Bessel transform, also known as Fourier Bessel transform or Hankel transform Window function Bessel window Bessel window Besselian date, see Epoch astronomy Besselian years MV Bessel , a German merchant ship in service 1928 45, latterly for the Kriegsmarine Bessel is the name of FriedrichBessel 1784 1846 , German mathematician, astronomer, and systematizer of the Bessel functions Bessel Kok born 1941 , Dutch businessman and chess organiser See also Bessell disambiguation Category Surnames Category Given names de Bessel Begriffskl rung fr Bessel nl Bessel ja sk Bessel sr ... more details
Unreferenced stub auto yes date December 2009 1552 Bessel alternate designation 1938 DE is a asteroid belt Main Belt asteroid . It was discovered on February 24, 1938 by Yrj V is l at Turku , Finland . It was named in honor of the German astronomer Friedrich Wilhelm Bessel . Minor planets navigator 1551 Argelander 1553 Bauersfelda Small Solar System bodies DEFAULTSORT Bessel Category Main Belt asteroids Category Asteroids named for people Category Discoveries by Yrj V is l Category Astronomical objects discovered in 1938 Beltasteroid stub de 1552 Bessel el 1552 eo 1552 Bessel eu 1552 Bessel fa it 1552 Bessel la 1552 Bessel hu 1552 Bessel ja no 1552 Bessel nn 1552 Bessel pl 1552 Bessel pt 1552 Bessel sk 1552 Bessel sr 1552 sv 1552 Bessel tl 1552 Bessel uk 1552 vi 1552 Bessel yo 1552 Bessel ... more details
lunar crater data image Image Bessel crater Apollo 15.jpg 200px caption Lunar crater Bessel from Apollo 15 . NASA photo. latitude 21.8 N or S N longitude 17.9 E or W E diameter 16 km depth 1.7 km colong 342 eponym Friedrich Wilhelm BesselFriedrich W. BesselBessel is a small moon lunar Impact crater crater that is located in the southern half of the Mare Serenitatis . Despite its small size, this is the largest crater to lie entirely within the Lunar mare mare . It lies to the north northeast of the crater Menelaus crater Menelaus . This crater is circular and bowl shaped with a rim that has a higher albedo than the floor or the surrounding mare. The outer rim is not significantly worn, and there are no features of note on the interior, apart from some slumping of material from the inner walls to the floor. Bessel is not of sufficient size to have developed the wiktionary terrace terrace structures of larger craters. It was named after Friedrich Wilhelm Bessel . A large Ray system ray , most likely from Tycho crater Tycho , crosses the mare from north to south, passing Bessel s western side. Satellite craters By convention these features are identified on lunar maps by placing the letter on the side of the crater mid point that is closest to Bessel. class wikitable width 25 style background eeeeee Bessel width 25 style background eeeeee Latitude width 25 style background eeeeee Longitude width 25 style background eeeeee Diameter align center D align center 27.3 N align center 19.9 E align center 5 km align center F align center 21.2 N align center 13.8 E align center 1 km align ... Astronomical Union IAU . Bessel A &mdash See Sarabhai crater . Bessel E &mdash See Bobillier crater ... title Radar Strip Showing Crater Bessel date November 23, 2009 publisher NASA url http www.nasa.gov ... craters on the Moon da Bessel m nekrater de Bessel Mondkrater fa fr Bessel crat re ru sk Bessel kr ter ... more details
Unreferenced date February 2012 The Bessel ellipsoid or Bessel 1841 is an important reference ellipsoid of geodesy . It is currently used by several countries for their national geodetic surveys, in Europe and on other continents, but will be replaced in the next decades by modern ellipsoids of satellite geodesy . The Bessel ellipsoid was derived 1841 by Friedrich Wilhelm Bessel , based on several meridian arc s and other data of continental geodetic network s of Europe , Russia and the British Survey of India . It is based on 10 meridional arcs and 38 precise measurements of the astro geographic latitude and longitude see also Astro geodetic astro geodesy . The dimensions of the ellipsoid axes were defined by logarithm s in keeping with former calculation methods. The Bessel and GPS ellipsoids The Bessel ellipsoid fits especially well to the geoid curvature of Europe and Eurasia . Therefore it is optimal for National survey networks in these regions, despite of the fact that its axes are about 700 m shorter than that of the mean Earth ellipsoid derived by satellites. Below the are the two axes a , b and the flattening a &minus b a . For comparison, the data of the modern World Geodetic System WGS84 are shown, which is mainly used for modern surveys and the Global Positioning System GPS system. Bessel ellipsoid 1841 defined by log a and a 6,377,397.155 ... data published by Bessel 1841 were then the best and most modern data mapping the figure of the Earth ... work of Bessel. In 1950 about 50 of the European triangulation networks and about 20 of other continent s networks were based on the Bessel ellipsoid. In the following decades the Americas ... satellite states in Eastern Europe to use the Krassowski ellipsoid of about 1940. As of 2010 the Bessel ... and Latitude degrees into UTM coordinates Category Geodesy de Bessel Ellipsoid fr Ellipso de de Bessel nl Ellipso de van Bessel ru sv Bessels ellipsoid ... more details
In mathematics , the Bessel potential is a potential theory potential named after Friedrich Wilhelm Bessel similar to the Riesz potential but with better decay properties at infinity. If s is a complex number with positive real part then the Bessel potential of order s is the operator math I Delta s 2 math where is the Laplace operator and the fractional calculus fractional power is defined using Fourier transforms. See also Riesz potential Fractional integration Sobolev space Fractional Schr dinger equation References eom id B b110420 title Bessel potential operator first R. last Duduchava Citation last1 Grafakos first1 Loukas title Modern Fourier analysis publisher Springer Verlag location Berlin, New York edition 2nd series Graduate Texts in Mathematics isbn 978 0 387 09433 5 doi 10.1007 978 0 387 09434 2 mr 2463316 year 2009 volume 250 eom id B b120170 title Bessel potential space first L.I. last Hedberg eom id B b015870 first E.D. last Solomentsev citation first Elias last Stein authorlink Elias Stein title Singular integrals and differentiability properties of functions publisher Princeton University Press location Princeton, NJ year 1970 isbn 0 691 08079 8 Category Fractional calculus Category Partial differential equations Category Potential theory Category Singular integrals ... more details
s name is a reference to FriedrichBessel , a German mathematician 1784 1846 , who developed the mathematical theory on which the filter is based. 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. ref ... filter3 hide In electronics and signal processing , a Bessel filter is a type of linear filter with a maximally flat group delay maximally linear phase response . Bessel filters are often used in audio crossover systems. Analog Bessel filters are characterized by almost constant group delay across ... order low pass Bessel filter. Note that the transition from the pass band to the stop band is much slower than for other filters, but the group delay is practically constant in the passband. The Bessel filter maximizes the flatness of the group delay curve at zero frequency. A Bessel low pass ...&dq Bessel filter polynomial&lr &as brr 3&ei gyeWSvTbIpmwNPyaqNcH v onepage&q Bessel 20filter 20polynomial ... Bessel polynomial from which the filter gets its name and math omega 0 math is a frequency chosen ... 0 math . Bessel polynomials File Bessel 3rd order poles.svg right thumb The roots of the third order Bessel polynomial are the pole zero plot poles of filter transfer function in the s plane s plane , here plotted as crosses. The transfer function of the Bessel filter is a rational function whose denominator is a reverse Bessel polynomial , such as the following math n 1 quad s 1 math math n 2 quad ... n 5 quad s 5 15s 4 105s 3 420s 2 945s 945 math The reverse Bessel polynomials are given by ref name ...,n. math Example File Bessel 3rd order gain.svg right thumb Gain plot of the third order Bessel filter, versus normalized frequency File Bessel 3rd order delay.svg right thumb Group delay plot of the third order Bessel filter, illustrating flat unit delay in the passband The transfer function for a third order three pole Bessel low pass filter , normalized to have unit group delay, is math H s frac ... more details
In mathematics , a Bessel process , named after FriedrichBessel , Why is it named after him? is a type of stochastic process . The Bessel process of order n is the real number real valued process X given by math X t W t , math where · denotes the Norm mathematics Euclidean norm Euclidean norm in R sup n sup and W is an n dimensional Wiener process Brownian motion started from the origin. The Bessel process of order n is the solution to the stochastic differential equation math dX t dZ t frac n 1 2 frac dt X t math where Z is a 1 dimensional Wiener process Brownian motion . Note that this SDE makes sense for any real parameter math n math although the drift term is singular at zero . Since W was assumed to have started from the origin the initial condition is X sub 0 sub 0. For n &ge 2, the n dimensional Wiener process is Markov chain Recurrence transient from its starting point almost surely with probability one , X sub t sub > 0 for all t > 0. It is, however, neighbourhood recurrent for n 2, meaning that with probability 1, for any r 0, there are arbitrarily large t with X sub t sub r on the other hand, it is truly transient for n 2, meaning that X sub t sub &ge r for all t sufficiently large. A notation for the Bessel process of order n started at zero is BES sub 0 sub n . References cite book author ksendal, Bernt title Stochastic Differential Equations An Introduction with Applications publisher Springer, Berlin year 2003 isbn 3 540 04758 1 Williams D. 1979 Diffusions, Markov Processes and Martingales, Volume 1 Foundations. Wiley. ISBN 0 471 99705 6. Category Stochastic processes probability stub ... more details
In mathematics, Bessel functions , first defined by the mathematician Daniel Bernoulli and generalized by FriedrichBessel , are Canonical form canonical solutions y x of Bessel s differential equation ... the order of the Bessel function the most common and important cases are for an integer or half ... different Bessel functions for these two orders e.g., so that the Bessel functions are mostly smooth functions of . Bessel functions are also known as cylinder functions or cylindrical harmonics ... of Bessel function 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 ... in cylindrical coordinate systems, one obtains Bessel functions of integer order n in spherical ... radiation Bessel functions also have useful properties for other problems, such as signal processing e.g., see FM synthesis , Kaiser window , or Bessel filter . Definitions Since this is a second order ... are described below. Bessel functions of the first kind J sub &alpha sub Bessel functions of the first kind, denoted as J sub sub x , are solutions of Bessel s differential equation that are finite ... x are defined by its Bessel function Properties properties below. It is possible to define the function ... function to non integer values. The graphs of Bessel functions look roughly like oscillating ... in terms of J sub n 1 sub x by the identities Bessel function Properties below . Image Bessel Functions 1st Kind, n 0,1,2 .svg thumb 300px right Plot of Bessel function of the first kind, J sub ... linearly independent solution is then found to be the Bessel function of the second kind, as discussed below. Bessel s integrals Another definition of the Bessel function, for integer values of math ... mathrm i , n tau x sin tau , mathrm d tau. math This was the approach that Bessel used, and from this definition ... to hypergeometric series The Bessel functions can be expressed in terms of the generalized hypergeometric ... more details
File Bessel beam.svg thumb Diagram of Axicon and resulting Bessel Beam File Bessek beam intensity.svg thumb Cross section of the Bessel beam and graph of intensity File Bessel beam reform.svg thumb Bessel beam re forming central bright area after obstruction A Bessel beam is a field of electromagnetic, acoustic or even gravitational radiation whose amplitude is described by a Bessel function of the first ... pmid 12226659 issue 6903 bibcode 2002Natur.419..145G ref ref D. McGloin, K. Dholakia, Bessel beams diffraction in a new light, Contemporary Physics 46 2005 15 28 ref A true Bessel beam is non diffractive ... spot. Bessel beams are also self healing , meaning that the beam can be partially obstructed at one point, but will re form at a point further down the beam axis . As with a plane wave a true Bessel ... because they exhibit little or no diffraction over a limited distance. Approximations to Bessel beams are made in practice by focusing a Gaussian beam with an axicon lens to generate a Bessel Gauss beam. The properties of Bessel beams ref F. O. Fahrbach, P. Simon, A. Rohrbach, Microscopy with self ... of an electromagnetic zero order Bessel beam by a dielectric sphere. Optics Letters 36 2011 766 768 ref make them extremely useful for Optical tweezers optical tweezing , as a narrow Bessel beam will maintain ... manipulation with acoustical tweezers may be feasible with a Bessel beam that scatters ref P. L. Marston, Scattering of a Bessel beam by a sphere, J. Acoust. Soc. Am. 121 2007 753 758 ref ref G. T. Silva, Off axis scattering of an ultrasound bessel beam by a sphere. IEEE Trans. Ultrason. Ferroelectr ... order Bessel vortex beam by a rigid sphere, Wave Motion 48 2011 392 400 ref and produces a radiation ... and quasi standing zero order Bessel beam tweezers, Annals of Physics 323 2008 1604 1620 ref ... and quasi standing zero order Bessel beam tweezers of variable half cone angles, IEEE Transactions ... acoustic radiation force of a high order Bessel beam on a rigid sphere, IEEE Transactions on Ultrasonics ... more details
Bessel Kok born Hilversum , 13 December 1941 is a Dutch business man and chess organiser living in Prague . He has served in top management positions in telecommunications companies in Belgium Belgacom and in the Czech Republic . He was also President of the Belgian based Banking communications company Society for Worldwide Interbank Financial Telecommunication SWIFT and responsible for SWIFT s sponsorship of several major International chess events. Kok was the Chairman of World Chess Grandmaster Association from 1985 until 1991, and helped to establish the 2002 Prague Agreement concerning the World Chess Championship . Kok is also active in the art s he was the producer of the movie film Nicholas Winton, the power of Good about British businessman Nicholas Winton who rescued children from the Nazi s. The film won an Emmy Award in 2002. Kok ran a campaign for presidency of FIDE the World Chess Federation FIDE against the incumbent Kirsan Ilyumzhinov in 2006. The election was held in June, 2006 in Torino , Italy during the World Chess Olympiad and Kok lost 54 96 to Ilyumzhinov. External links http www.fide.com news.asp?id 960 Two Tickets Announced for FIDE Elections http www.chessville.com Editorials Interviews 20Questions Kok.htm Chessville Interviews 20 Questions with Bessel Kok Persondata Metadata see Wikipedia Persondata . NAME Kok, Bessel ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH 13 December 1941 PLACE OF BIRTH DATE OF DEATH PLACE OF DEATH DEFAULTSORT Kok, Bessel Category 1941 births Category Living people Category Dutch chess players Category Dutch businesspeople Category People from Hilversum Category Chess officials Category Belgacom Group Netherlands business bio stub netherlands chess bio stub ca Bessel Kok de Bessel Kok fr Bessel Kok nl Bessel Kok fi Bessel Kok ... more details
In statistics , Bessel s correction , named after FriedrichBessel , is the use of n 1 instead of n in the formula for the sample variance and sample standard deviation , where n is the number of observations in a sample it corrects the bias in the estimation of the population variance, and some but not all of the bias in the estimation of the population standard deviation. That is, when estimation theory estimating the population variance and standard deviation from a sample when the population mean is unknown, the sample variance is a biased estimator of the population variance, and systematically underestimates it. Multiplying the standard sample variance by n n 1 equivalently, using 1 n 1 instead of 1 n in the estimator s formula corrects for this, and gives an unbiased estimator of the population variance. The cost of this correction is that the unbiased estimator has uniformly higher mean squared error than the biased estimator. In some terminology, ref W.J. Reichmann, W.J. 1961 Use and abuse of statistics , Methuen. Reprinted 1964 1970 by Pelican. Appendix 8. ref ref Upton, G. Cook, I. 2008 Oxford Dictionary of Statistics , OUP. ISBN 978 0 19 954145 4 entry for Variance data ref the factor n n 1 is itself called Bessel s correction . A subtle point is that, while the sample variance using Bessel s correction is an unbiased estimate of the population variance, its square root , the sample standard deviation, is a biased ..., such as the normal see unbiased estimation of standard deviation for details. One can understand Bessel ... i.e. without Bessel s correction s sup 2 sup is the unbiased sample variance i.e. with Bessel s correction ... i 1 n x i right 2 n 1 n left frac n n 1 right ,s n 2 math Proof that Bessel s correction yields an unbiased ... Bessel s Correction DEFAULTSORT Bessel s Correction Category Statistical deviation and dispersion Category Statistical inference hu Bessel f le korrekci ... more details
In mathematics , especially functional analysis , Bessel s inequality is a statement about the coefficients of an element math x math in a Hilbert space with respect to an orthonormal sequence . Let math H math be a Hilbert space, and suppose that math e 1, e 2, ... math is an orthonormal sequence in math H math . Then, for any math x math in math H math one has math sum k 1 infty left vert left langle x,e k right rangle right vert 2 le left Vert x right Vert 2 math where , denotes the inner product space inner product in the Hilbert space math H math . If we define the infinite sum math x sum k 1 infty left langle x,e k right rangle e k, math consisting of infinite sum of vector resolute math x math in direction math e k math , Bessel s inequality mathematics inequality tells us that this series mathematics series Limit of a sequence converges . One can think of it that there exists math x in H math which can be described in terms of potential basis math e 1, e 2, ... math . For a complete orthonormal sequence that is, for an orthonormal sequence which is a Orthonormal basis basis , we have Parseval s identity , which replaces the inequality with an equality and consequently math x math with math x math . Bessel s inequality follows from the identity math 0 le left x sum k 1 n langle x, e k rangle e k right 2 x 2 2 sum k 1 n langle x, e k rangle 2 sum k 1 n langle x, e k rangle 2 x 2 sum k 1 n langle x, e k rangle 2, math which holds for any natural n . planetmath title Bessel inequality id 3089 See also Cauchy Schwarz inequality External links http mathworld.wolfram.com BesselsInequality.html Bessel s Inequality the article on Bessel s Inequality on MathWorld. Category Hilbert space Category Inequalities ca Desigualtat de Bessel de Besselsche Ungleichung es Desigualdad de Bessel eo Neegala o de Bessel fr In galit de Bessel it Disuguaglianza di Bessel kk hu Bessel egyenl tlens g ro Inegalitatea lui Bessel ru fi Besselin ep yht l ... more details
Citations missing article date January 2008 Vasily Vasil yevich Bessel lang ru April 25 OS April 13 , 1843 1842? St Petersburg March 1, OS February 16 1907, Zurich was a Russian music publisher. Bessel graduated from the Saint Petersburg Conservatory in 1865 studying violin with Henryk Wieniawski and viola with Hieronymus Weickmann . He co founded the thriving publishing firm V. Bessel and Co. since 1869 and a print shop since 1871 , which published works by prominent Russian composers, notably Pyotr Tchaikovsky , Anton Rubinstein , Alexander Dargomyzhsky and the members of the New Russian Musical School Modest Musorgsky , Nikolai Rimsky Korsakov Nikolay Rimsky Korsakov , C sar Cui , Mily Balakirev , and Alexander Borodin . He was the journal s publisher and also the sole editor of a weekly St Petersburg magazine Muzykal ny listok The Musical Leaf from September 3, 1872 to June 5, 1877, that appeared on Sundays during the nine month Russian musical season, from September October to May June. The journal s purpose was to offer an in depth view of the many aspects of Russian and foreign musical life. Later he published the journal Muzykal noye obozrenie The musical revue 1885 1888 . He also wrote a book Notnoe delo The Note Matter , published in 1901. His brother N. V. Bessel was a co owner of the firm V. Bessel and Co. External links http www.ripm.org journal info.php?ABB MUL Muzykal ny listok Persondata Metadata see Wikipedia Persondata . NAME Bessel, Vasily ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH 1843 PLACE OF BIRTH DATE OF DEATH 1907 PLACE OF DEATH DEFAULTSORT Bessel, Vasily Category Russian editors Category 1843 births Category 1907 deaths ru , ... more details
Location map Greenland lat dir N lat deg 75 lat min 59 lon dir W lon deg 21 lon min 10 width 300 caption Location within Greenland label Bessel Fjord Bessel Fjord is a fjord in Erik the Red s Land in northeastern Greenland . It is located in the Greenland Caledonites. ref cite book url http books.google.co.uk books?id 0YhkhnIuJwgC&pg PA172&lpg PA172&dq Bessel Fjord&source bl&ots IWfb4ow2ku&sig s jye6hc 7JEJZTVzv8vSJTK6Y&hl en&ei dNVATNyoCJKy0gTQ7OmgDw&sa X&oi book result&ct result&resnum 6&ved 0CCoQ6AEwBQ v onepage&q Bessel 20Fjord&f false title The Greenland Caledonides evolution of the northeast margin of Laurentia authors A. K. Higgins, Jane A. Gilotti, M. Paul Smith work Volume 202 of Memoir Geological Society of America publisher Geological Society of America year 2008 page 172 isbn 0813712025 ref References Reflist coord 75 59 N 21 10 W scale 1000000 display title Category Fjords of Greenland Category Article Feedback 5 greenland geo stub ... more details
More footnotes date September 2009 In mathematics , the Bessel polynomials are an orthogonal polynomials ..., is sometimes known as the reverse Bessel polynomials See Grosswald 1978, Berg 2000 . math theta ... are the same as the first but in reverse order. For example, the third degree Bessel polynomial is math y 3 x 15x 3 15x 2 6x 1 , math while the third degree reverse Bessel polynomial is math theta 3 x x 3 6x 2 15x 15 , math The reverse Bessel polynomial is used in the design of Bessel filter Bessel electronic filters . Properties Definition in terms of Bessel functions The Bessel polynomial may also be defined using Bessel function s from which the polynomial draws its name. math y n x ... frac 2 pi x ,e 1 x K n frac 1 2 1 x math where math K n x math is a modified Bessel function of the second ... as a hypergeometric function The Bessel polynomial may also be defined as a confluent hypergeometric ... left frac 2 x right n 1 U left n 1,2n 2, frac 2 x right . math The reverse Bessel polynomial may ... The Bessel polynomials have the generating function math sum n 0 sqrt frac 2 pi x n frac 1 2 e x K n frac 1 2 x frac t n n e x 1 sqrt 1 2t . math Recursion The Bessel polynomial may also be defined ... 2 theta n 2 x , math Differential equation The Bessel polynomial obeys the following differential equation ... of the Bessel polynomials have been suggested in literature Krall, Fink , as following math y n ... math a curve surrounding the 0 point. They specialize to the Bessel polynomials for math alpha beta 2 math , in which situation math rho x e 2 x math . Rodrigues formula for Bessel polynomials The Rodrigues formula for the Bessel polynomials as particular solutions of the above differential equation ... 2n e frac beta x math where math a n alpha, beta math are normalization coefficients. Associated Bessel polynomials According to this generalization we have the following generalized associated Bessel ... authorlink Leonard Carlitz coauthors year 1957 month title A Note on the Bessel Polynomials journal ... more details
Ship propulsion Screw propellor Ship speed Ship capacity Ship crew Ship notes Ship armament Bessel .... She was sold in 1926 and renamed Bessel . She was seized by the Allies in Vigo , Spain , in May 1945 ... to Otwi Werke GmbH , Bremen and renamed Bessel . In December 1928, ref name Leokragh she was sold ... publisher Plimsoll Ship Data accessdate 24 June 2010 ref In 1940, Bessel was requisitioned by the Kriegsmarine ... Ships Bessel was used to refuel U boat s eight times during the war. Although supposedly a merchant ... of Bessel during the Second World War http www.photoship.co.uk JAlbum 20Ships Old 20Ships 20Ba slides Birgitte 20Skou 01.html Photo of Birgitte Skou Empire C ships DEFAULTSORT Bessel Category 1925 ... more details
In mathematical analysis , the Bessel Clifford function , named after FriedrichBessel and William Kingdon Clifford , is an entire function of two complex variable s that can be used to provide an alternative development of the theory of Bessel function s. If math pi x frac 1 Pi x frac 1 Gamma x 1 math is the entire function defined by means of the reciprocal Gamma function , then the Bessel Clifford ... for all regions with bounded z , and hence the Bessel Clifford function is an entire function of the two complex variables n and z . Differential equation of the Bessel Clifford function ... xy n 1 y y. qquad math This equation is of generalized hypergeometric type, and in fact the Bessel ... the hypergeometric function being normalized so that its value at z 0 is one. Relation to Bessel functions The Bessel function of the first kind can be defined in terms of the Bessel Clifford function ... we can see from this that the Bessel function is not entire. Similarly, the modified Bessel function ... 2 4 right . math The procedure can of course be reversed, so that we may define the Bessel Clifford ... C n x , math which defines the recurrence relationship for the Bessel Clifford function. This is equivalent ... . math It can be shown that this continued fraction converges in all cases. The Bessel Clifford function of the second kind The Bessel Clifford differential equation math xy n 1 y y qquad math has ... mathcal K math correspond to the Bessel functions of the second kind. We have math K n x left frac ... Hence just as the Bessel function and modified Bessel function of the first kind can both be expressed ... William Kingdon title On Bessel s Functions journal Mathematical Papers location London year 1882 pages 346 349 . Citation first A. George last Greenhill title The Bessel Clifford function, and its ... 232 242 . Citation first G. N. last Watson authorlink G. N. Watson title A Treatise on the Theory of Bessel ... . DEFAULTSORT Bessel Clifford Function Category Complex analysis Category Special hypergeometric ... more details
V. Bessel and Co. was a musical firm founded in 1869 in St Petersburg by Vasily Bessel Vasily Vasil yevich Bessel 1843 1907 . His brother N. V. Bessel was a co owner of the firm. The firm and a print shop since 1871 published works by prominent Russian composers, notably Pyotr Tchaikovsky , Anton Rubinstein , Alexander Dargomyzhsky and the members of the New Russian Musical School Modest Musorgsky , Nikolai Rimsky Korsakov Nikolay Rimsky Korsakov , C sar Cui , Mily Balakirev , and Alexander Borodin . The firm issued a weekly magazine Muzykal ny listok The Musical Leaf from September 3, 1872 to June 5, 1877, and Muzykal noye obozrenie The musical revue 1885 1888 . External links http www.ripm.org journal info.php?ABB MUL Muzykal ny listok Category Music publishing companies Bessel Category Opera publishers Bessel Category Publishing companies established in 1869 ... more details
Friedrich may refer to Names Friedrich surname , people with the surname FriedrichFriedrich given name , people with the given name Friedrich Other Friedrich board game , a board game about Frederick the Great and the Seven Years War Friedrich novel Friedrich novel , a novel about anti semitism written by Hans Peter Richter Friedrich Air Conditioning , a company manufacturing air conditioning and purifying products SS Friedrich , a German cargo ship in service 1941 45 See also Frederick disambiguation Nikolaus Friedreich disambig als Friedrich Begriffskl rung de Friedrich Begriffskl rung et Friedrich es Friedrich fr Friedrich ko nl Friedrich no Friedrich pl Friedrich ru sv Friedrich ... more details
Wikify date September 2011 In mathematics, the Bessel Maitland function , or Wright generalized Bessel function , is a generalization of the Bessel function , introduced by harvs txt authorlink Edward Maitland Wright first Edward Maitland last Wright year 1934 . The word Maitland in the name of the function seems to be the result of confusing Edward Maitland Wright s middle and last names. It is given by math J mu, nu z sum k ge 0 frac z k Gamma k mu nu 1 k . math References Citation last1 Wright first1 E. M. title The asymptotic expansion of the generalized Bessel function. doi 10.1112 plms s2 38.1.257 jfm 60.0306.02 year 1934 Category Special functions ... more details
In mathematics, a Jackson q Bessel function or basic Bessel function is one of the three q analog s of the Bessel function introduced by harvs txt authorlink F. H. Jackson last Jackson year1 1903 year2 1903b year3 1905 year4 1905b . The third Jackson q Bessel function is the same as the Hahn Exton q Bessel function . Definition The three Jackson q Bessel functions are given in terms of the Pochhammer symbol and the basic hypergeometric function by math J nu 1 x q frac q nu 1 q infty q q infty x 2 nu 2 phi 1 0,0 q nu 1 q, x 2 4 math math J nu 2 x q frac q nu 1 q infty q q infty x 2 nu 0 phi 1 q nu 1 q, x 2q nu 1 4 math math J nu 3 x q frac q nu 1 q infty q q infty x 2 nu 1 phi 1 0 q nu 1 q,qx 2 4 math References Citation last1 Ismail first1 Mourad E. H. title The zeros of basic Bessel functions, the functions J sub ax sub x , and associated orthogonal polynomials doi 10.1016 0022 247X 82 90248 7 mr 649849 year 1982 journal Journal of Mathematical Analysis and Applications issn 0022 247X volume 86 issue 1 pages 1 19 Citation last1 Jackson first1 F. H. title On generalized functions of Legendre and Bessel year 1903 journal Trans. Roy. Soc. Edinburgh volume 41 pages 1 28 Citation last1 Jackson first1 F. H. title Theorems relating to a generalization of the Bessel functions year 1903 journal Trans. Roy. Soc. Edinburgh volume 41 pages 105 118 Citation last1 Jackson first1 F. H. title Theorems relating to a generalization of Bessel s function. jfm 36.0513.02 year 1904 journal Edinb. Roy. Soc. Trans. volume 41 pages 399 408 Citation last1 Jackson first1 F. H. title The Application of Basic Numbers to Bessel s and Legendre s Functions doi 10.1112 plms s2 2.1.192 year 1905 journal Proc. London Math. Soc. volume 2 issue 1 pages 192 220 Citation last1 Jackson first1 F. H. title The Application of Basic Numbers to Bessel s and Legendre s Functions Second paper doi 10.1112 plms s2 3.1.1 year 1905 journal Proc. London Math. Soc. volume 3 issue 1 pages 1 23 Category Special functions ... more details
In mathematics , Fourier Bessel series is a particular kind of generalized Fourier series an infinite series expansion on a finite interval based on Bessel function s. Fourier Bessel series are used in the solution to partial differential equation s, particularly in cylindrical coordinate systems. Definition The Fourier Bessel series may be thought of as a Fourier expansion in the coordinate of cylindrical coordinates . Just as the Fourier series is defined for a finite interval and has a counterpart, the continuous Fourier transform over an infinite interval, so the Fourier Bessel series has a counterpart over an infinite interval, namely the Hankel transform . Because Bessel function s are orthogonal with respect to a weight function math x math on the interval math 0,b math , they can be expanded in a Fourier Bessel series defined by math f x sim sum n 0 infty c n J alpha lambda n x b math , where math lambda n math is the n th zero of math J alpha x math . This series is associated with the boundary condition math f b 0 math . From the orthogonality relationship math int 0 1 J alpha x lambda m ,J alpha x lambda n ,x ,dx frac delta mn 2 J alpha 1 lambda n 2 math , the coefficients are given by math c n frac int 0 b J alpha lambda n x b ,f x ,x ,dx int 0 b x J alpha 2 lambda n x b dx frac langle f, J alpha lambda n x b rangle J alpha lambda n x b 2 . math The lower integral may be evaluated, yielding math c n frac int 0 b J alpha lambda n x b ,f x ,x ,dx b 2 J alpha pm 1 2 lambda n 2 math , where the plus or minus sign is equally valid. Dini series A second Fourier Bessel series, also known as Dini series , is associated with the Robin boundary condition math b f b c f b 0 math , where math c math is an arbitrary constant. The Dini series can be defined by math f x sim ... cite web last Weisstein first Eric. W authorlink Eric W. Weisstein title Fourier Bessel Series work ... Bessel series applied to Acoustic Field analysis on http www.trinnov.com en about us research overview ... more details
Johann Franz Bessel in religion Gottfried b. 5 September 1672, at Buchen , in the Grand Duchy of Baden d. at G ttweig Abbey G ttweig , 22 January 1749 was a German Benedictine abbot and historian. Life He made his course in the humanities at Aschaffenburg , W rzburg , and Bamberg , and in 1690 entered the University of Salzburg , conducted by the Benedictines, where he specialized in philosophy, also attending lectures on theology and jurisprudence. He entered the Benedictine Order at G ttweig on the Danube , Lower Austria , 15 June 1692. After making his vows 21 June 1693 , he completed his theological course at Vienna , was ordained 21 March 1696 , and on 23 May was granted the degree of Doctor of Theology, being shortly afterwards appointed Lector in philosophy and theology in the monastery of Seligenstadt on the River Main . In 1699 he was summoned to the electoral court of Mainz by Archbishop Lothar Franz von Schonborn , who immediately sent him to Rome to study the curial practice of the Rota Romana . Having completed a two years course in law, he obtained the degree of Doctor Juris Utriusque , and on his return to Mainz 1703 he was appointed vicar general and supreme judge of the ecclesiastical court of the archdiocese of Mainz . He was also employed on various diplomatic missions, as, for instance, to the court of Brunswick Wolfenb ttel in connection with the conversion ... of Vienna. Abbot Bessel was the second founder of G ttweig, which became, under his rule of thirty ... scale. Personally, Abbot Bessel was a retiring religious, presenting to all a shining example of monastic .... References Albert, Gottfried Bessel und das Chronicon Gottwicense in Freiburger Diocesan Archiv. 1899 ... article Catholic Persondata Metadata see Wikipedia Persondata . NAME Bessel, Johann Franz ALTERNATIVE ... 1749 PLACE OF DEATH G ttweig Abbey DEFAULTSORT Bessel, Johann Franz Category 1672 births Category ... Bessel ... more details
DISPLAYTITLE q Bessel polynomials In mathematics, the q Bessel polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme . harvs txt last1 Koekoek first1 Roelof last2 Lesky first2 Peter A. last3 Swarttouw first3 Ren F. title Hypergeometric orthogonal polynomials and their q analogues publisher Springer Verlag location Berlin, New York series Springer Monographs in Mathematics isbn 978 3 642 05013 8 doi 10.1007 978 3 642 05014 5 mr 2656096 year 2010 loc 14 give a detailed list of their properties. Definition The polynomials are given in terms of basic hypergeometric function s and the Pochhammer symbol by math displaystyle math Orthogonality Empty section date September 2011 Recurrence and difference relations Empty section date September 2011 Rodrigues formula Empty section date September 2011 Generating function Empty section date September 2011 Relation to other polynomials Empty section date September 2011 References Citation last1 Gasper first1 George last2 Rahman first2 Mizan title Basic hypergeometric series publisher Cambridge University Press edition 2nd series Encyclopedia of Mathematics and its Applications isbn 978 0 521 83357 8 doi 10.2277 0521833574 mr 2128719 year 2004 volume 96 Citation last1 Koekoek first1 Roelof last2 Lesky first2 Peter A. last3 Swarttouw first3 Ren F. title Hypergeometric orthogonal polynomials and their q analogues publisher Springer Verlag location Berlin, New York series Springer Monographs in Mathematics isbn 978 3 642 05013 8 doi 10.1007 978 3 642 05014 5 mr 2656096 year 2010 dlmf id 18 title first Tom H. last Koornwinder first2 Roderick S. C. last2 Wong first3 Roelof last3 Koekoek first4 Ren F. last4 Swarttouw Category Orthogonal polynomials Category Q analogs Category Special hypergeometric functions ... more details
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