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Topic: Bessel function

Related Topics

Hankel function

Bessel differential equation

List of functions

Gamma function

Wronskian

Kaiser window

Laurent series

Cylindrical coordinates

Handbook of Mathematical Functions

Gaussian elimination

Normal distribution

Friedrich Bessel

Frequency spectrum

FM synthesis

Casio CZ synthesizers

	Bessel function - Wikipedia, the free encyclopedia
		Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates, and Bessel functions are therefore especially important for many problems of wave propagation, static potentials, and so on.
		Bessel functions also have useful properties for other problems, such as signal processing (e.g., see FM synthesis or Kaiser window).
		The Bessel functions are valid even for complex arguments x, and an important special case is that of a purely imaginary argument.
	en.wikipedia.org /wiki/Bessel_function (1445 words)

	Friedrich Bessel - Wikipedia, the free encyclopedia
		Bessel was the son of a civil servant, and at the age of 14 he was apprenticed to the import-export concern Kulenkamp.
		With this work under his belt, Bessel was able to achieve the feat for which he is best remembered today: he is credited with being the first to use parallax in calculating the distance to a star.
		In 1838 Bessel won the "race", announcing that 61 Cygni had a parallax of 0.314 arcseconds; which, given the diameter of the Earth's orbit, indicated that the star was ~3 parsecs away.
	en.wikipedia.org /wiki/Friedrich_Bessel (550 words)

	BESSEL FUNCTION - LoveToKnow Article on BESSEL FUNCTION (Site not responding. Last check: 2007-11-06)
		The symbol generally adopted to represent these functions is J,,, (p) where m denotes the order of the function.
		He discussed their properties and constructed tables for their evaluation Although Bessel was the first to systematically treat of these functions, it is to be noted that in 1732 Daniel Bernoulli obtained the function of zero order as a solution to the problem of the oscillations of a chain suspended at one end.
		In 1764 Leonhard Euler employed the functions of both zero and integral orders in an analysis into, the vibrations of a stretched membrane; an investigation which has been considerably developed by Lord Rayleigh, who has also shown (1878) that Bessels functions are particular cases of Laplaces functions.
	81.1911encyclopedia.org /B/BE/BESSEL_FUNCTION.htm (317 words)

	Bessel Functions
		An important class of functions can be formed using only the four processes of arithmetic, called the rational functions, and these constitute a connection between the general hazy concept of function, and the concrete realization of functions in terms of numbers.
		Bessel functions are of this type, solutions of certain differential equations that arise in many different connections.
		Cylinder Functions, and they turn out to be the Bessel Functions because the satisfy the differential equation, but he carefully points out that the converse must be proved, and cannot be assumed from one particular case.
	www.du.edu /~jcalvert/math/bessels.htm (1849 words)

	Friedrich Bessel: biography and encyclopedia article (Site not responding. Last check: 2007-11-06)
		Bessel was a contemporary of Carl Gauss[Follow this hyperlink for a summary of this subject], EHandler: no quick summary.
		Bessel was the son of a civil servant, EHandler: no quick summary.
		(Bessel was able to achieve the feat for which he is best remembered today: he is credited with being the first to use parallax parallax quick summary:
	www.absoluteastronomy.com /encyclopedia/f/fr/friedrich_bessel.htm (1949 words)

	Math Is Fun Forum / Please help me on Bessel Function
		A Bessel function (of the first kind, I mean here) can nowdays be routinely calculated if given the order of the Bessel function.
		Bessel Function plots are also known corresponding to specific orders.
		I know nothing about Bessel functions, but I went to the address you gave, and the three graphs shown there all look different--they all intersect the y-axis in different spots.
	www.mathsisfun.com /forum/viewtopic.php?id=1778 (402 words)

	[No title]
		They satisfy Bessel's equation: 2 2 2 x y'' + x y' + (x - v) y = 0 - BesselI and BesselK are the modified Bessel functions of the first and second kinds, respectively.
		They satisfy the modified Bessel equation: 2 2 2 x y'' + x y' - (x + v) y = 0 - HankelH1 and HankelH2 are the Hankel functions, also known as the Bessel functions of the third kind.
		Function: AiryAi, AiryBi - The Airy wave functions Calling Sequence: AiryAi(x) AiryBi(x) AiryAi(n,x) AiryBi(n,x) Parameters: x - an algebraic expression n - an algebraic expression, assumed to be a non-negative integer Description: - The Airy wave functions AiryAi and AiryBi are linearly independent solutions for w in the equation w''-x*w=0.
	www.math.niu.edu /~rusin/known-math/99/mod_bessel (711 words)

	Mathematical Methods Special Functions Bessel’s Equation and Bessel Functions (in more detail)
		These functions are applicable in a wide variety of situations and, similar to the other special functions, one particular set of Bessel functions also has the property of orthogonality.
		It is important to have a feeling for the functional behavior of the Bessel functions for various values of the argument x.
		The are many more useful relationships for the Bessel functions that have not been tabulated here, and the student is encouraged to browse the literature for a more comprehensive treatise on this subject.
	gershwin.ens.fr /vdaniel/Doc-Locale/Cours-Mirrored/Methodes-Maths/white/math/s8/s8bess/s8bess.html (1067 words)

	Bessel function
		Sometimes the Bessel function is called the cylinder function.
		However, the Fourier transform of this 3d function, is a Bessel function of the first kind (of first order).
		The Bessel function of the third kind or Hankel function H
	www.2dcurves.com /gamma/gammabs.html (326 words)

	Bessel functions (Site not responding. Last check: 2007-11-06)
		Bessel functions made their first appearance by relating the angular position of a planet moving along a Keplerian ellipse to elapsed time.
		However, the radial functions in the Schrödinger equation are Laguerre polynomials, and the one dimensional Schrödinger equation for a constant force are Airy functions which can be transformed into Bessel functions of order 1/3.
		It would hardly be possible not to include a sample of Jahnke and Emde's drawings in a discussion of Bessel Functions, such as the one which can be seen in Figure 16.
	delta.cs.cinvestav.mx /~mcintosh/comun/complex/node57.html (356 words)

	Maxima Manual: 16. Special Functions (Site not responding. Last check: 2007-11-06)
		Controls expansion of the Bessel functions when the order is half of an odd integer.
		This function is thus a canonical simplifier for sums of powers of sine and cosine terms of a particular type.
		are functions of two arguments which are a coefficient and a trigonometric part of a term in series respectively.
	maxima.sourceforge.net /docs/manual/en/maxima_16.html (970 words)

	FRIEDRICH WILHELM BESSEL - LoveToKnow Article on FRIEDRICH WILHELM BESSEL (Site not responding. Last check: 2007-11-06)
		Placed at the age of fifteen in a counting-house at Bremen, he was impelled by Ins desire to obtain a situation as supercargo on a foreign voyage to study navigation, mathematics and finally astronomy.
		Apart from the large scope of his activity, he introduced such important novelties as the effective use of the heliometer, the correction for personal equation (in 1823), and the systematic investigation of instrumental errors.
		In pure mathematics he enlarged the resources of analysis by the invention of Bessels Functions.
	85.1911encyclopedia.org /B/BE/BESSEL_FRIEDRICH_WILHELM.htm (523 words)

	besselh (MATLAB Functions)
		Bessel function of the third kind (Hankel function)
		The relationship between the Hankel and Bessel functions is
		If one input is a row vector and the other is a column vector, the result is a two-dimensional table of function values.
	www-rohan.sdsu.edu /doc/matlab/techdoc/ref/besselh.html (197 words)

	spherical bessel function
		When complex arguement (eg/ 10+j 13) for sperical bessel functions in MATHCAD is used the result is very large(10 exp 22).
		The spherical Bessel function (aka half-order Bessel functions) is found in the case of systems with spherical symmetry.
		Like Bessel functions there are 3 kinds of Spherical Bessel functions- first kind, second kind and spherical hankel functions.
	www.edaboard.com /ftopic83285.html (160 words)

	BESSEL FUNCTION - Online Information article about BESSEL FUNCTION
		symbol generally adopted to represent these functions is Jm (p) where m denotes the order of the function.
		He discussed their properties and constructed tables for their evaluation Although Bessel was the first to systematically treat of these functions, it is to be noted that in 1732 See also:
		Heine has shown that the functions of higher orders may be considered as limiting values of the associated functions; this relation was discussed independently, in 1878, by Lord Rayleigh.
	encyclopedia.jrank.org /BER_BLA/BESSEL_FUNCTION.html (564 words)

	Bessel Function Suite for MATLAB (Site not responding. Last check: 2007-11-06)
		The core of the suite is a Fortran MEX file based on a Bessel function library written by D. Amos at Sandia National Laboratory several years ago.
		Bessel functions of complex argument are fully supported.
		The primary functions in the suite are: BESSELJ(NU,Z) Bessel functions of the first kind, BESSELY(NU,Z) Bessel functions of the second kind, BESSELI(NU,Z) Modified Bessel functions of the first kind, BESSELK(NU,Z) Modified Bessel functions of the second kind, BESSELH(NU,K,Z) Hankel functions, AIRY(K,Z) Airy functions.
	www.csc.fi /math_topics/Mail/NANET94/msg00253.html (141 words)

	Class jnt.functions.Bessel
		Evaluates the hyperbolic Bessel function of the first kind of order 1 for a given real argument x.
		Evaluates the Bessel function of the first kind of order 0 for a given real argument x.
		Evaluates the Bessel function of the first kind of order 1 for a given real argument x.
	math.nist.gov /jnt/api/jnt.functions.Bessel.html (382 words)

	: Class JSpecfun
		Y = ERFC(X) is the complementary error function for each element of X. The complementary error function is defined as: erfc(x) = 2/sqrt(pi) * integral from x to inf of exp(-t^2) dt.
		The scaled complementary error function is defined as: erfcx(x) = exp(x^2) * erfc(x) which is approximately (1/sqrt(pi)) * 1/x for large x.
		The gamma function is defined as: gamma(x) = integral from 0 to inf of t^(x-1) exp(-t) dt.
	www.cs.utexas.edu /~kane/Java_Docs_HTML/jamlab/JSpecfun.html (1122 words)

	Bessel Function Table (Site not responding. Last check: 2007-11-06)
		In general, a function is part of an answer to a question about why some object or process occurred in a system thatevolved or was designed with some goal.
		Thus function refers forward from the object or process, along some chain of causes tothe goal or evolutionary success.
		The common feature of a function is that it produces some kind of result.
	www.daikaiju.com /edge/23837-besselfunctiontable.html (446 words)

	Bessel (Scalable Simulation Framework)
		Returns the Bessel function of the first kind of order 0 of the argument.
		Returns the Bessel function of the first kind of order 1 of the argument.
		Returns the modified Bessel function of the third kind of order 1 of the argument.
	www.ssfnet.org /javadoc/cern/jet/math/Bessel.html (784 words)

	BER
		Compute the ber Kelvin function or the derivative of the ber Kelvin function.
		The functions described here compute these functions for the case when there is a real, non-negative input argument and order zero (i.e., v = 0).
		Functions can accept (and return) either parameters (i.e., single values) or variables (i.e., an array of values) while let subcommands are specific in which they accept as input and what they return as output.
	www.itl.nist.gov /div898/software/dataplot/refman2/auxillar/ber.htm (468 words)

	Bessel Function
		Since Bessel's differential equation is a second order ordinary differential equation, two sets of functions, the Bessel function of the first kind
		Recurrence Relation: A Bessel function of higher order can be expressed by Bessel functions of lower orders.
		is a piecewise continuous function, generally the non-homogeneous term of the problem.
	www.efunda.com /math/bessel/bessel.cfm (352 words)

	Modified Bessel Function
		Similar to the relations between the trigonometric functions and the hyperbolic trigonometric functions,
		Generating Function: The generating function of the Bessel Function of the first kind is
		Recurrence Relation: A modified Bessel function of higher order can be expressed by modified Bessel functions of lower orders.
	www.efunda.com /math/bessel/modifiedbessel.cfm (95 words)

	GNU Scientific Library -- Reference Manual - Special Functions (Site not responding. Last check: 2007-11-06)
		The special functions are available in two calling conventions, a natural form which returns the numerical value of the function and an error-handling form which returns an error code.
		The Debye functions are defined by the integral D_n(x) = n/x^n \int_0^x dt (t^n/(e^t - 1)).
		The transport functions J(n,x) are defined by the integral representations J(n,x) := \int_0^x dt t^n e^t /(e^t - 1)^2.
	www.lsw.uni-heidelberg.de /manuals/gsl-ref-html/gsl-ref_7.html (5988 words)

	Bessel Function Zeroes - Finch (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		93 A Treatise on the Theory of Bessel Functions (context) - Watson - 1944
		1 Asymptotic expansion for zeros of Bessel functions and of th..
		1 a conjecture regarding the extrema of Bessel functions and i..
	citeseer.ist.psu.edu /679069.html (512 words)

	Colt 1.0.3 - API Specification: Class Bessel (Site not responding. Last check: 2007-11-06)
		Returns the modified Bessel function of order 0 of the argument.
		Returns the exponentially scaled modified Bessel function of order 0 of the argument.
		Returns the modified Bessel function of order 1 of the argument.
	hoschek.home.cern.ch /hoschek/colt/V1.0.3/doc/cern/jet/math/Bessel.html (688 words)

	Bessel Functions
		On new families of summation formulae of ordinary and generalized Bessel functions.
		Bessel functions of two variables: some power series and plots.
		Singular differential operators with r-1 parameters, and Bessel functions of a vector index.
	math.fullerton.edu /mathews/n2003/beselfunction/BesselFunctionBib/Links/BesselFunctionBib_lnk_3.html (2096 words)

	besselj (MATLAB Functions)
		is a real constant, is called Bessel's equation, and its solutions are known as Bessel functions.
		The Bessel functions are related to the Hankel functions, also called Bessel functions of the third kind,
		function uses a Fortran MEX-file to call a library developed by D. Amos [3] [4].
	www.technion.ac.il /guides/matlab/techdoc/ref/besselj.html (290 words)

	J0 - maths special bessel j - C / C++ and Excel Numerical Component
		Bessel function of the first kind, with order zero and exponential scaling.
		Returns modified Bessel function of the first kind, with order zero.
		In the second interval, the Hankel asymptotic expansion is employed with two rational functions of degree 6/6 and 7/7.
	www.codecogs.com /d-ox/maths/special/bessel/j/j0.php (179 words)

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