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Topic: Hankel function

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	Bessel function - Wikipedia, the free encyclopedia
		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.
		The Bessel functions are valid even for complex arguments x, and an important special case is that of a purely imaginary argument.
		In fact, there are simple closed-form expressions for the Bessel functions of half-integer order in terms of the standard trigonometric functions, and therefore for the spherical Bessel functions.
	en.wikipedia.org /wiki/Bessel_function (1217 words)

	Clearing up the market cycle... best Spherical Hankel Function... (Site not responding. Last check: 2007-10-14)
		Periodia in one's mathematical form appeals to this function, but it is fold form from value of price row, values of time, characteristic for given row, function sine and parameter of optimization called as Chimerical price.
		Spherical Hankel Function of the Second Kind -- from MathWorld Spherical Hankel Function of the Second Kind -- from MathWorld h_n^{(2)}(x) \equiv \sqrt{\pi\over 2x} H_{n+1/2}^{(2)}(x) = j_n(x)-in_n(x), where H^{(2)}(x) is the Hankel function...
		Function of the Second Kind, Spherical Hankel Function of the First Kind, Spherical Hankel Function of the Second Kind, Spherical...
	ascot.pl /th/Fourier5/Spherical-Hankel-Function....htm (501 words)

	[No title]
		Be careful though that the recurrence relations for this modified Hankel's functions are somewhat different from those of the normal Bessel and Hankel's functions.
		I got myself into a fix while evaluating the derivatives of the modified Hankel's functions because there was a mistake in one of the books I used.
		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)

	Operator Theory in Unit Disk (Site not responding. Last check: 2007-10-14)
		The Berezin transform of a function in D is the Bergman space analog of the Poisson transform of a function on the circle.
		In the case of real-valued symbols, the boundedness and compactness problem of Hankel operators was first considered by Zhu in his Ph.D. thesis [37] and then completely settled by Bekkole, Berger, Coburn, and Zhu in [7].
		The general boundedness and compactness of a Hankel operator on the Bergman space was finally settled by Luecking in [28].
	math.albany.edu:8000 /~kzhu/operator.html (639 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)

	[No title]
		This function is thus a canonical simplifier for sums of powers of SIN's and COS's of a particular type.
		In particular, all elliptic functions and integrals use the parameter m instead of the modulus k or the modular angle \alpha.
		The xi are merely dummy variables as in function definition headers and are used to indicate the ith argument of f.
	www.mit.edu:8001 /afs/athena.mit.edu/software/maxima/info/maxima.info-7 (4934 words)

	Maxima Manual - Orthogonal Polynomials (Site not responding. Last check: 2007-10-14)
		The following table lists each function in specfun, its Maxima name, restrictions on its arguments (m and n must be integers), and a reference to the algorithm specfun uses to evaluate it.
		For some functions, when the order is symbolic but has been declared to be an integer, specfun will return a series representation.
		Many functions in specfun are computed as a special case of the Jacobi polynomials; they also enjoy the speed boost from the modedeclared version of jacobi.
	www.ma.utexas.edu /maxima/maxima_16.html (1215 words)

	HANKEL Function
		The HANKEL function generates a Hankel matrix from a vector, or a block Hankel matrix from a matrix.
		A block Hankel matrix has the property that all matrices on the reverse diagonals are the same.
		The HANKEL function is especially useful in time-series applications, where the covariance matrix of a set of variables representing the present and past and a set of variables representing the present and future is often assumed to be a block Hankel matrix.
	www.asu.edu /sas/sasdoc/sashtml/iml/chap17/sect113.htm (319 words)

	Spectral Analysis of Surface Waves Software (Site not responding. Last check: 2007-10-14)
		Since the plane wave solution is used, the methods begin to exhibit increasing bias as the wavelength increases relative to the array length, but calculation efficiency is gained through use of the fft.
		The method determines an optimum vector of weights as a function of frequency, and therefore the filter does not change structure as a function of wavenumber.
		The methods use the correct Hankel function solution to the cylindrical wave equation, allowing the power spectrum to be estimated for longer wavelengths relative to the array length.
	www.zywicki.com /software.htm (645 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		The square root and the complex exponential functions are available in most programming languages, although in some cases it could be beneficial (in terms of computational cost) not to use the standard language functions.
		In the two-dimensional case, operation M requires the computation of the spherical Hankel functions or log functions when k is zero.
		Since Hankel functions are not generally available as standard functions, then some external routine is required for their evaluation.
	www.boundary-element-method.com /acoustics/manual/chap3/sect3_4.htm (360 words)

	Bessel Function
		is a piecewise continuous function, generally the non-homogeneous term of the problem.
		The transform based on this relationship is called a Hankel Transform.
		Modified Bessel Function: Similar to the relations between the trigonometric functions and the hyperbolic trigonometric functions,
	www.efunda.com /math/bessel/bessel.cfm (352 words)

	GNU Scientific Library -- Reference Manual - Discrete Hankel Transforms
		Specifically, let f(t) be a function on the unit interval.
		The implementation allows a scaling of the fundamental interval, for convenience, so that one can assume the function is defined on the interval [0,X], rather than the unit interval.
		This function returns the value of the n'th sample point in the unit interval, (j_{\nu,n+1}/j_{\nu,M}) X. These are the points where the function f(t) is assumed to be sampled.
	www.inference.phy.cam.ac.uk /pjc51/local/gsl/manual/gsl-ref_30.html (330 words)

	Maybe this can forecast the market trend... best Hankel Function (Site not responding. Last check: 2007-10-14)
		Hankel Function: Similar to, the Hankel function of the first kind...
		Hankel Function -- from MathWorld Hankel Function -- from MathWorld There are two types of functions known as Hankel functions.
		Hankel Function of the Second Kind -- from MathWorld Hankel Function of the Second Kind -- from MathWorld H_n^{(2)}(z) \equiv J_n(z)-iY_n(z), where J_n(z) is a Bessel function of the first kind and Y_n(z) is a Bessel function of the second...
	www.ascot.pl /th/Fourier4/Hankel-Function.htm (529 words)

	FFTLog
		This singularity reflects a real singularity in the corresponding continuous Hankel transform (unlike the singularity of the previous paragraph, which is an avoidable artefact of discreteness).
		From the definition (2) of the continuous Hankel transform, it can be seen that periodically replicating a function a(r) in logarithmic space lnr and then taking its continuous Hankel transform is equivalent to Hankel transforming the function a(r) and then periodically replicating the transform ã(k) in lnk.
		It may help to bias the function with a power law before transforming it, as in the second algorithm in §7.
	casa.colorado.edu /~ajsh/FFTLog (3446 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		To: bug-octave at che dot utexas dot edu Date: Mon, 21 Aug 95 09:51:36 MDT I was asked to post a patched version of hankel.m, regarding my bug reports.
		This perhaps isn't the most efficient approach, but this version of hankel will generate similar output to Matlab's hankel function (zeros below main anti-diag if only one arg is passed), and it contains only a few minor adjustments from the original version.
		------------------------------------------------------------------------- function retval = hankel (c, r) # usage: hankel (c, r) # # Return the Hankel matrix constructed given the first column # c, and (optionally) the last row r.
	www.octave.org /mailing-lists/bug-octave/1995/281 (252 words)

	spherical bessel function (Site not responding. Last check: 2007-10-14)
		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 (196 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		The Hankel function $H_{\nu}^{(1)}(z)$ is one of the third kind of Bessel function.
		The Hankel function $H_{\nu}^{(2)}(z)$ is one of the third kind of Bessel function.
		The hypergeometric function $F(a, b; c; z)$ is the solution of the hypergeometric differential equation $z(1-z)w^{\prime\prime}+[c-(a+b+1)z]w^{\prime}-abw=0$.
	www.csd.uwo.ca /faculty/watt/home/research/openmath/spfun1.ocd (3083 words)

	Bessel function (Site not responding. Last check: 2007-10-14)
		The functions are found for systems with cylindrical symmetry.
		The Bessel function of the third kind or Hankel function H
		(x) is a (complex) combination of the two solutions: the real part is the Bessel function of the first kind, the complex part the Bessel function of the second kind.
	www.2dcurves.com /gamma/gammabs.html (245 words)

	Chapter IV
		4.19.1 The Bessel function and its integral representation:
		We shall rather apply this instead of to the Hankel function H immediately to the Neumann function (33) in which we are mainly interested for the determination of the singularity of concern.
		In contrast, the Bessel functions J (as well as the Neumann functions N) becomes infinite asymptotically in both half planes.
	mpec.sc.mahidol.ac.th /radok/physmath/mat10/41.htm (4453 words)

	Plane and Spherical Wave Spectra
		For incoming waves, the conjugate spherical Hankel function is used.
		For an expansion that is finite everywhere, the spherical Bessel function replaces the spherical Hankel function.
		The index n is restricted to integer values because the Legendre function would otherwise be infinite on the z-axis, and m must be an integer for the field to be a continuous function of
	www.silcom.com /~aludwig/Physics/Exact_piston/Wave_spectra.htm (1316 words)

	Notes 9+10
		We are solving the particle in a finite "square" well in 3-D (really, it's a spherical well, of course) We found the form of the radial wave function, but haven't yet used continuity of R(r) and R'(r) at the point r=a.
		This means that far away, the wave function really looks about the same, whether there's a potential well there or not.
		As we argued before, this wave function should not change when phi increases by 2pi, which implies m must be an integer.
	www.colorado.edu /physics/phys3220/3220_fa97/notes/notes9_10/notes9_10_7.html (1084 words)

	The Hyperbolic Geometry of SAR Imaging
		This problem is primarily a geometric one, that of accounting for curved wavefronts: the spirit of this paper is to emphasize these geometrical properties over electromagnetic ones.
		Hankel transforms are less familiar than Fourier transforms but have similar properties.
		Note that I have made no approximations to the geometry of the situation, I only made an approximation to the Hankel function to speed up the data processing.
	home.cfl.rr.com /pt/Hyper.htm (3509 words)

	Chapter IV
		We will, as a rule, prefer the representation by Hankel functions to that by Bessel or Neumann functions, especially our complex integrals yield for those an equally convenient expression as for these.
		-plane, whereby one has to imagine that in the representations of the two H functions, because of the branching at the end of the section C, these must be cut along a suitable half-ray.
		The special suitability of the Hankel functions for problems of damped vibrations rest on this.
	kr.cs.ait.ac.th /~radok/math/mat/41.htm (4441 words)

	Optimal Hankel-Norm Approximation and Rational Function Models Using Impulse Response Data - Storming Media
		Abstract: The motivation of this work is rational function modeling of transfer functions using impulse response data from underwater acoustic transducers.
		However, since such data does not, in general, correspond to a rational model even in the abscence of noise, an effective rational function approximation criterion is necessary.
		The main contribution of this report is to extend Kung's model reduction algorithm to approximating systems not exactly described by rational function models of finite order and to requiring only the solution of simple instead of generalized eigenvalue problems required in Kung's method.
	www.stormingmedia.us /71/7131/A713132.html (291 words)

	List of letters used in mathematics and science - Wikipedia, the free encyclopedia
		H - Auxiliary magnetic field, Enthalpy, Hamiltonian, Hankel function, Heaviside step function, Henry, Higgs boson, Homology group, Hubble constant, Hydrogen, Set of Quaternions,
		Y - Bessel function of the second kind, SI prefix: (yotta-), Yttrium,
		Z - Atomic number, Impedance, Set of integers, Partition function (statistical mechanics), Partition function (quantum field theory), Z boson, SI prefix: (zetta-),
	en.wikipedia.org /wiki/List_of_letters_used_in_mathematics_and_science (661 words)

	sci.math.num-analysis: Re: Integrating Bessel fcns, kelvin fcns, hankel functions
		Re: Integrating Bessel fcns, kelvin fcns, hankel functions
		(or a kelvin or hankel function for that matter)...
		first kind of Hankel function that is used, but that is right,...
	sci.tech-archive.net /Archive/sci.math.num-analysis/2004-07/0222.html (275 words)

	The semiclassical Green function (Site not responding. Last check: 2007-10-14)
		The amplitude of the Green function is the square root of the intensity
		For the semiclassical expression of the Green function in presence of curved walls the effective radius is given [11] by the product of the distance between the source and the first bounce point
		The stretching factor is the analog of the optical magnification factor of curved mirrors.
	galahad.elte.hu /~vattay/cserti/node3.html (419 words)

	Bio Energy Hankel - The best energy products, sites and information on the web today! (Site not responding. Last check: 2007-10-14)
		calculus of residues, gamma function, Hankel functions, modified Bessel functions...
		Zeros of Hankel Functions and Poles of Scattering Amplitudes, (with...
		Laplace, Fourier, Hankel, and Mellin transforms, with applications to the...
	energy.jaydirs.com /index.php?k=bio-energy-hankel (750 words)

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