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	PlanetMath: Laplace equation in cylindrical coordinates
		The differential equations of (3) and (5) are ordinary differential equations, while (6) is a little more complicated and we must turn to Bessel functions.
		Delving into all the nuances of solving Bessel's differential equation is beyond the scope of this article, however, the curious are directed to Watson's in depth treatise [Watson].
		This is version 6 of Laplace equation in cylindrical coordinates, born on 2006-11-14, modified 2006-11-14.
	planetmath.org /encyclopedia/LaplaceEquationInCylindricalCoordinates.html (813 words)

	NationMaster - Encyclopedia: Bessel function (Site not responding. Last check: 2007-10-13)
		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.
		The Bessel functions can be expressed in terms of the hypergeometric series as In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k.
		Another important property of Bessel's equations, which follows from Abel's identity, involves the Wronskian of the solutions: In mathematics, Abels identity (also called Abels differential equation identity) is an equation that expresses the Wronskian of two homogeneous solutions of a second-order linear ordinary differential equation in terms of the coefficients of the original differential equation.
	www.nationmaster.com /encyclopedia/Bessel-function (3153 words)

	Bessel functions
		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)

	Kids.Net.Au - Encyclopedia > Bessel function (Site not responding. Last check: 2007-10-13)
		Bessel functions, invented by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation:
		The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportional to 1/√x (see also their asymptotic forms, below), although their roots are not generally periodic except asymptotically for large x.
		In this case, the solutions to the Bessel equation are called the modified Bessel functions of the first and second kind, and are defined by:
	www.kids.net.au /encyclopedia-wiki/be/Bessel_function (1026 words)

	Bessel Functions
		Differential equations effectively define new functions, called their solutions, and it is the task of the mathematician to elucidate their properties so that they can be used with effect and confidence.
		Bessel functions are of this type, solutions of certain differential equations that arise in many different connections.
		Bessel himself used the integral, which first arose in a problem in celestial mechanics, to investigate his functions.
	www.du.edu /~jcalvert/math/bessels.htm (1849 words)

	[No title]
		In an ordinary differential equation we are usually trying to solve for a function, y(x), where the equation involves derivatives of y with respect to x.
		Equations [1] and [3] are first-order differential equations.
		Applying equation [106] to this problem we have the interval from 0 to R (in place of a to b) and we have a real function so we do not have to consider the complex conjugate.
	www.csun.edu /~lcaretto/me501a/ODE2.doc (7127 words)

	Bessel function
		Bessel was a mathematician who lived from 1784 to 1846.
		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).
	www.2dcurves.com /gamma/gammabs.html (326 words)

	besselh (MATLAB Function Reference) (Site not responding. Last check: 2007-10-13)
		Bessel functions of the third kind (Hankel functions)
		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.cs.berkeley.edu /titan/sww/software/matlab/techdoc/ref/besselh.html (125 words)

	PlanetMath: Bessel's equation
		is a non-negative constant, is called the Bessel's equation.
		The similar definition is set for the first kind Bessel function of an arbitrary order
		This is version 3 of Bessel's equation, born on 2007-01-16, modified 2007-01-16.
	planetmath.org /encyclopedia/BesselEquation.html (263 words)

	Bessel Functions
		One of the varieties of special functions which are encountered in the solution of physical problems is the class of functions called Bessel functions.
		The solutions to this equation are in the form of infinite series which are called Bessel funtions of the first kind.
		Bessel functions are encountered in physical situations where there is cylindrical symmetry.
	hyperphysics.phy-astr.gsu.edu /HBASE/math/bessel.html (173 words)

	Drumhead Vibration Lab (Site not responding. Last check: 2007-10-13)
		The Matlab assignment below asks you to plot in 3-D and animate the motion of a circular drumhead, based upon equations for the radial and circumferential nodal patterns..
		Equation 2 is in the form of a Helmholtz equation.
		The R-side of the equation becomes Bessel's equation, whose solutions R(r) are transcendental functions called Bessel functions of the first kind, J
	fubini.swarthmore.edu /~ceverba1/Drum.html (953 words)

	31 (Site not responding. Last check: 2007-10-13)
		We then began our discussion of the modified Bessel's equation which emerges as the radial equation, upon separation of variables, for pdes involving the laplacian in polar coordinates.
		Thus Bessel's equation is not of Sturm-Liouville form; we won't be able to invoke Sturm-Liouville results to obtain orthogonality conditions for this equation.
		Such rescaling of coefficients in differential equations by rescaling a spatial argument is a common analysis technique, especially in numerical work, to express computations in terms of dimensionless quantities.
	www.physics.emory.edu /faculty/benson/320/notes/31/31.html (332 words)

	The Radius of Convergence of Series Solutions
		The first question could be answered by finding the radius of convergence of the power series, but it turns out that there is an elegant Theorem, due to Lazarus Fuchs (1833-1902), which solves both of these questions simultaneously.
		As it turns out, Bessel's Equation does indeed not always have solutions, which can be written as power series.
		If you are interested in Bessel's Equation, look up the section on "The Method of Frobenius" in a differential equations or advanced engineering mathematics textbook.
	www.sosmath.com /diffeq/series/series05/series05.html (469 words)

	FAQTs - Knowledge Base - View Entry - Math:Calculus:Differential equation:Partial:Linear:Order:2: Can you show ...
		Chief among these are the Bessel functions, a knowledge of which shall be necessary for the solution of for example the following problem: 'A solid (actually a cylinder verticaly cutted in two halves) consists of a right circular cylinder of radius b and height h.
		Because of the nature of the boundaries of the solid, it will be highly INconvenient to use the heat equation in the cartesian form, that is: [1] ((second partial derivative of temperature to x) plus (second partial derivative of temperature to y) plus (second partial derivative of temperature to z)) equals ((constant2).
		Now this is the usual trigonometric equation (think about the graph of a sine function, which keeps on cutting the x-axis), which has as a solution a multiple of 180 degrees.
	faqts.com /knowledge_base/view.phtml/aid/35542 (1453 words)

	The Skin effect
		This is Helmholz' equation, or Bessel's differential equation with n=0, and solutions are the Bessel functions J0 and Y0.
		The Bessel functions are part of libm (#include ), and are there named j0, j1, jn.
		For higher values of betaR the results from the Bessel's* integral (N=9999), are used to adapt a polynomial of degree 8 to two or three ranges of betaR. Well, actually it is a polynomial of 1/R as function of 1/(betaR), that way it works a lot better.
	home.swipnet.se /swi/bessel/bessel.html (1472 words)

	[No title]
		Subject: Re: Modified Bessel Function: a name Date: Wed, 29 Dec 1999 07:10:22 -0500 Newsgroups: sci.math.num-analysis Pawel F. Gora wrote: > During some calculations I came across the function K_n(z), a > solution to the modified Bessel equation.
		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.
	www.math.niu.edu /~rusin/known-math/99/mod_bessel (711 words)

	Particles in Two-Dimensional Boxes
		We learned from solving Schrödinger's equation for a particle in a one-dimensional box that there is a set of solutions, the stationary states, for which the time dependence is just an overall rotating phase factor, and these solutions correspond to definite values of the energy.
		The solutions to this equation are determined by the boundary conditions on psi, in general there is a sequence of such eigenstates labeled by a quantum number n = 0,1,2,3,…, with corresponding values E
		The equation for the radial function R(r) is actually a well known equation in mathematical physics, it is Bessel's equation.
	galileo.phys.virginia.edu /classes/252/2d_wells.html (867 words)

	“Nondiffracting” Light Beam
		showed that Bessel functions could be used to create exact solutions to the free space wave equation.
		The Bessel beam solution results in a beam profile that has a narrow central area with concentric rings surrounding it.
		Figure 2 Intensity distribution of a finite energy Bessel beam, which is proportional to the square of the zeroth-order Bessel function.
	www.u.arizona.edu /~sergioj/report.html (965 words)

	Loss-less propagation
		To obtain the pressure distribution along the axis of the cylinder equation (2.52) was substituted into equation (2.33).
		This was multiplied by the transverse eigenfunction from equation (2.48) to give the full pressure field.
		The real value of the complex pressure amplitude was chosen so that a snapshot of the pressure field is shown, rather than a time averaged value, as would be the case if the absolute value was shown.
	www.ph.ed.ac.uk /~jonathan/thesis/node24.html (571 words)

	Special Functions
		Bessel functions arise in solving differential equations for systems with cylindrical symmetry.
		are the Bessel functions, error function, incomplete gamma function, and Hermite and Laguerre polynomials.
		Bessel functions of the first kind can be expressed in terms of the
	documents.wolfram.com /v5/TheMathematicaBook/AdvancedMathematicsInMathematica/MathematicalFunctions/3.2.10.html (1399 words)

	besselj, bessely (MATLAB Function Reference)
		J = besselj(nu,Z) Bessel function of the 1st kind
		Y = bessely(nu,Z) Bessel function of the 2nd kind
		The Bessel functions are related to the Hankel functions, also called Bessel functions of the third kind,
	www.cs.berkeley.edu /titan/sww/software/matlab/techdoc/ref/besselj.html (323 words)

	X. PDEs on a disk
		This equation would apply to the equilibrium temperature distribution in a thin disk insulated on its two flat faces, or to a long cylinder along the z-axis, provided that the temperature does not depend on z.
		Note that if we were to consider the wave equation rather than the heat equation, with Dirichlet boundary conditions on a circle, the spatial dependence of the normal modes would be the same as the eigenfunctions of the Laplace operator as found in Model Problem X.2.
		Suppose that the ends of a cylinder of length L and radius A are insulated and the curved side is held at temperature 0.
	www.mathphysics.com /pde/ch10wr.html (2120 words)

	Bessel Functions
		The goal of this section is to summarize a few results on Bessel functions.
		is a half integer, then it is natural to view the Bessel function of the first kind as a continuous generalization of the trigonometric functions.
		The polylogarithm is first defined on the unit disc, and then extended to the complex plane through a variety of functional equations and integral representations.
	www.math.ubc.ca /~matrogers/bessel/bessel.html (375 words)

	powerseriesI.htm
		A second motivation for the use of power series techniques lies in the realm of partial differential equations and a fundamental method for solving them, separation of variables.
		Substituting in the equation and simplifing leads to two ordinary differential equations.
		The bottom line here is that power series techniques give a powerful approach to the study of solutions of Bessel's equation and many others arising in mathematical physics.
	germain.umemat.maine.edu /faculty/bray/Archive_notes/powerseriesI.htm (1293 words)

	DSolve:: Bessel's differential equation
		Hi buddies, Here is a differential equation, which could not be done by DSolve (in Version 5.0).
		This occurs in the mathematical modelling of a simple pendulum of length L and a parameter k.
		(1) The above equation is in fact reducible to Bessel's differential equation (with order n = 0) with the substituions L-x = z and s = 2 k Sqrt[z] y''[s] + 1 /s y'[s] + y[s] ==0....
	www.codecomments.com /message796976.html (805 words)

	A New Look at the Airy Equation with Fences and Funnels
		Courses on ``methods of mathematical physics'' or ``special functions'' are largely the study of such equations: Legendre's equation, Bessel's equation, Airy's equation, the hypergeometric equation and Hermite's equation, to name a few.
		We will illustrate these methods for the Airy equation, where all the problems mentioned above occur, but the transformations and geometric methods can be used for most of the classical equations, and many other equations as well.
		Antifunnels for the Prüfer transformation of the Airy equation.
	www.cecm.sfu.ca /organics/papers/hubbard/paper/ohtml/paper.html (459 words)

	Bessel Function
		In the Sturm-Liouville Boundary Value Problem, there is an important special case called Bessel's Differential Equation which arises in numerous problems, especially in polar and cylindrical coordinates.
		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.
	www.efunda.com /math/bessel/bessel.cfm (352 words)

	XI. Great Balls of PDEs
		The colatitude is the angle measured down from the pole rather than up from the equator, which would be the usual geographic coordinate of latitude.
		This equation has some features of the multidimensional problem of Chapter IX and some of those of the problems on disks of Chapter X.
		The eigenvalue equation for u boils down to an ordinary differential equation for the function R: Equation (11.15) should remind you of Bessel's equation; the only difference is the factor of 2 in the second term.
	www.mathphysics.com /pde/ch11wr.html (2637 words)

	S.O.S. Mathematics CyberBoard :: View topic - Question about Bessel Functions
		This doesn't seem any simpler than the original differential equation (all we've done is changed one of the coefficients)...
		satisfies the differential equation at that point, provided that the function and its first two derivatives are finite at r = 0...
		My calc textbook has the bessel equation in it, although it's only one question, and we never covered the frobenius method in lectures.
	www.sosmath.com /CBB/viewtopic.php?p=96410 (881 words)

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