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	Analytic continuation - Wikipedia, the free encyclopedia
		In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function.
		Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined makes no good sense.
		That is because the difference is an analytic function which vanishes on the intersection of their domains, a non-empty open set, and an analytic function which vanishes on a non-empty open set must vanish everywhere on its domain (assuming the domain is connected) and hence must be identically zero.
	en.wikipedia.org /wiki/Analytic_continuation (1109 words)

	PlanetMath: analytic continuation
		The reason that the notion of analytic continuation is interesting is the rigidity theorem for complex functions, which implies that analytic continuation is unique.
		It is a simple consequence of the rigidity theorem that the result of analytically continuing a function along a path is independent of the decomposition of the path into arcs.
		This is version 7 of analytic continuation, born on 2004-10-03, modified 2006-02-24.
	planetmath.org /encyclopedia/AnalyticContinuation.html (456 words)

	Continuation - Wikipedia, the free encyclopedia
		See analytic continuation for the use of the term in complex analysis; see Continuation War for the Finno-Soviet conflict during World War II; and see continuation application for the special type of patent application.
		Continuations are also used in models of computation including the Actor model, process calculi, and the lambda calculus.
		Continuations are the functional expression of the GOTO statement, and the same caveats apply.
	en.wikipedia.org /wiki/Continuation (767 words)

	Analytic - Wikipedia, the free encyclopedia
		Analytic geometry, the study of geometry using the principles of algebra
		Analytic continuation, a technique to extend the domain of definition of a given analytic function
		Analytical Thomism, the movement to present the thought of Thomas Aquinas in the style of modern analytic philosophy
	en.wikipedia.org /wiki/Analytic (443 words)

	MAT 119 -- Honors Calculus (II) (Site not responding. Last check: 2007-10-21)
		Likewise, an analytic function is determined by its restriction to a very small neighborhood of any point within its domain of definition.
		If two analytic functions agree in some small neighborhood of a point, then it follows that they agree everywhere that both are defined.
		Or if two analytic functions agree in some small neighborhood of a point but the domain of one is smaller than the domain of the other, then it is sensible to think of the other as determining an "analytic contination" of the other.
	math.albany.edu:8000 /~hammond/course/series/ancon-ss.html (351 words)

	Riemann zeta function - Wikipedia, the free encyclopedia
		Bernhard Riemann realized that the zeta-function can be extended by analytic continuation in a unique way to a meromorphic function ζ(s) defined for all complex numbers s with s ≠ 1.
		This formula is used to construct the analytic continuation in the first place.
		Jonathan Sondow, "Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series", Proc.
	en.wikipedia.org /wiki/Riemann_zeta_function (1711 words)

	Continuation: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-21)
		A continuing patent application is a patent application which follows an "original" patent application....
		A programming language supports re-entrant or first-class continuations if a continuation may be invoked repeatedly to re-enter the same context.
		If a continuation may only be used to escape the current context to a surrounding one, EHandler: no quick summary.
	www.absoluteastronomy.com /encyclopedia/c/co/continuation.htm (1501 words)

	ipedia.com: Analytic continuation Article (Site not responding. Last check: 2007-10-21)
		Suppose f is an analytic function defined on the open...
		In practice, this continuation is typically done by first establishing some functional equation on the small domain and then using this equation to extend the domain.
		The concept of a universal cover was developed to define a natural domain for the analytic continuation of an analytic function.
	www.ipedia.com /analytic_continuation.html (293 words)

	the monodromy principle
		If all the coefficients of a Taylor's series vanish at a point of analyticity, and the radius of its disk of convergence is non-zero, then the function not only vanishes throughout that disk, but everywhere else that can be reached by analytic continuation.
		is an analytic function whenever f is, in spite of the complex conjugations.
		Thus values from the upper half plane can be reflected into the lower half plane and taken as an analytic continuation of the original function.
	delta.cs.cinvestav.mx /~mcintosh/comun/complex/node30.html (342 words)

	Math Forum Discussions (Site not responding. Last check: 2007-10-21)
		continuation" is not really the term you want here.
		analytic function equal to f(n) for all positive integers n.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /kb/thread.jspa?threadID=1312033&messageID=4143447 (201 words)

	The Analytic Impulse (Site not responding. Last check: 2007-10-21)
		A complex analytic function is to its real part as a solid object is to its shadow.
		The analytic impulse DELTA(t) is the analytic continuation of the familiar Dirac symbol delta(t).
		The complex (or quadrature) sum of the impulse response with its Hilbert transform is called the analytic impulse response (AIR) of the system, and the energy-time curve (ETC) is the magnitude of the AIR.
	www.andrewduncan.ws /AIR/AIR_0.html (723 words)

	The Analytic Continuation of the HyperPower function
		The principal branch of W, on the other hand, is analytic at 0, with radius of convergence R = 1/e.
		It is also interesting to note that D is nothing more than the region of analyticity of the principal branch of the Lambert's W, under the map exp(z), also as expected (!).
		Having seen the analyticity of F(z), it is natural now to want to use F(z) as the analytic continuation of the real hyperpower function F(x).
	ioannis.virtualcomposer2000.com /math/hyperpower.html (1077 words)

	Analytic Continuation In Representation Theory And Harmonic Analysis - Olafsson (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		Analytic Continuation In Representation Theory And Harmonic Analysis (2000)
		16 Analytic continuation of the holomorphic discrete series of..
		2 Analytic continuation of local representations of symmetric..
	citeseer.ist.psu.edu /324623.html (906 words)

	Analytic Functions and Singularities (Site not responding. Last check: 2007-10-21)
		Polynomials, ratios of polynomials, exponential and trigonometric functions are analytic except at points where the denominator goes to zero.
		If f is analytic in a region, it may be possible to extend f beyond this region so it stays analytic.
		If f is analytic in a region except for poles, it is called homomorphic.
	www.ee.cooper.edu /courses/course_pages/97_Fall/EE114/complex/node2.html (274 words)

	[No title]
		But it is a fundamental feature of analytic functions that the only (entire) one which vanishes on a nonempty open set is the zero function.
		Think of it this way: one way to define C^\infty functions is that they _have_ a Taylor series at each point; one way to define analytic functions is that on a neighborhood of each point this Taylor series actually _converges_ to the original function.
		You do need some analytic continuation to get across the critical strip, although for real values of s between 0 and 1 you may use the convergent series Sum((-1)^n/n^s) to define a function which is easily related to zeta.
	www.math.niu.edu /~rusin/known-math/00_incoming/zeta (951 words)

	Analytic continuation - (Site not responding. Last check: 2007-10-21)
		The continuation technique may, however, come up against inconsistencies (defining more than one value), or global obstructions.
		That is because the difference is an analytic function vanishing on a non-empty open set, and hence must be identically zero.
		$\mathcal G$ is sometimes called the universal analytic function.
	psychcentral.com /psypsych/Analytic_continuation (1114 words)

	CJO - Abstract
		Because of the finite-time singularity, it is impossible to regard the rolling-up spiral as a solution of the Birkhoff–Rott equation as long as time is real.
		However, it may be possible to analytically continue the equation to the spiral along a path to get around the singularity in complex-time plane.
		Distribution of the complex singularities and their limiting behaviour indicate that it is absolutely impossible to perform analytic continuation in complex-time domain to the spiral solution.
	dx.doi.org /10.1017/S0956792503005230 (255 words)

	Analytic Continuation
		Analytic continuation is carried out by expanding a function of
		The analytic continuation of the domain of Eq.
		Analytic continuation works for any finite number of poles of
	www-ccrma.stanford.edu /~jos/filters/Analytic_Continuation.html (288 words)

	EN224: LINEAR ELASTICITY (Site not responding. Last check: 2007-10-21)
		Some of the most interesting boundary value problems in linear elasticity have been solved using the idea of analytic continuation, which reduces many boundary value problems to a so-called Hilbert problem, with a known solution.
		The idea of analytic continuation provides a powerful tool for solving half-plane problems, and can also be used to solve problems involving regions with circular boundaries.
		There is a systematic approach you can follow to devise an appropriate continuation, however, which we will illustrate for the case of a traction boundary value problem.
	www.engin.brown.edu /courses/en224/complexcnt/complexcnt1.html (826 words)

	Math Forum Discussions (Site not responding. Last check: 2007-10-21)
		I don't know why everyone here is so against analytic functions that give certain sequences - in this case prime numbers.
		An easy way I can think of, would be doing an inverse mellin transform on the function P(s) (prime zeta function), which is given by a series in terms of the mobius function and logs of the zeta function.
		This would probably give an "analytic" function that gives pi(x) at x natural, and who knows what at anything else.
	mathforum.org /kb/thread.jspa?threadID=1156138&messageID=3788326 (128 words)

	Social Choice with Analytic Preferences
		Arrow's axioms for social welfare functions are shown to be inconsistent when the set of alternatives is the nonnegative orthant in a multidimensional Euclidean space and preferences are assumed to be either the set of analytic classical economic preferences or the set of Euclidean spatial preferences.
		When either of these preference domains is combined with an agenda domain consisting of compact sets with nonempty interiors, strengthened versions of the Arrovian social choice correspondence axioms are shown to be consistent.
		To help establish the economic possibility theorem, an ordinal version of the Analytic Continuation Principle is developed.
	ideas.repec.org /p/van/wpaper/0023.html (506 words)

	MathLinks Math Forum :: View topic - analytic continuation and the riemann function? (Site not responding. Last check: 2007-10-21)
		There are several ways to make an analytic continuation of this function to the entire complex plane by making it a mermomorphic function with a pole of residue 1 at z = 1 and no other singularities.
		Anyways, I remember rewriting the "normal" sum-definition as a sum of a convergent sum for Re s > 0 or mabye it was >= 0 and an integral then you repeadily can use the functional equation etc etc. I think this should work, don't have time to check it.
		But a search on "analytic continuation of zeta function" on google would give answers.
	www.mathlinks.ro /Forum/post-369.html (365 words)

	Amazon.com: Analytic Extension Formulas and their Applications (International Society for Analysis, Applications and ... (Site not responding. Last check: 2007-10-21)
		Analytic Extension is a mysteriously beautiful property of analytic functions.
		With this point of view in mind the related survey papers were gathered from various fields in analysis such as integral transforms, reproducing kernels, operator inequalities, Cauchy transform, partial differential equations, inverse problems, Riemann surfaces, Euler-Maclaurin summation formulas, several complex variables, scattering theory, sampling theory, and analytic number theory, to name a few.
		Research and graduate text featuring survey papers on analytic extension, from the point of view that analytic extensions are a beautiful property of analytic functions.
	www.amazon.com /exec/obidos/tg/detail/-/0792369505?v=glance (661 words)

	Properties of the Vacuum.1. Mechanical and Thermodynamic (1983)
		This Appendix derives formulae for analytic continuation and limiting behaviour of mode sums used in Sections 2 and 3.
		This form provides an analytic continuation for all values of
		An analytic continuation for multi-dimensional mode sums is obtained in direct analogy with (A.5) using the generalized Jacobi
	www.stephenwolfram.com /publications/articles/particle/83-properties1/10/text.html (164 words)

	Power series (Site not responding. Last check: 2007-10-21)
		This expansion converges in the right-hand circle, which overlaps with the unit circle in the shaded region.
		This is called analytic continuation, and in this way one can cover the whole of the complex plane, with the exception of singularities, by convergent Taylor series expansions of the function.
		is analytic, and which can be reached from the original domain without crossing singularities.
	www.astro.cf.ac.uk /undergrad/module/PX3211/com/node4.html (273 words)

	Flattening and Analytic Continuation (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		We use this to produce an example of an a#noid morphism that cannot be flattened by a finite sequence of local blow-ups.
		Thus the global rigid analogue, [7], Theorem 2.3, of Hironaka's complex analytic flattening theorem is not true.
		1 A systematic approach to rigid analytic geometry (context) - Bosch, Guntzer et al.
	citeseer.ist.psu.edu /601287.html (471 words)

	Analytic continuation for asymptotically AdS 3D gravity (Site not responding. Last check: 2007-10-21)
		We have previously proposed that asymptotically AdS 3D wormholes and fl holes can be analytically continued to the Euclidean signature.
		The analytic continuation procedure was described for non-rotating spacetimes, for which a plane t = 0 of time symmetry exists.
		In the present paper we generalize this analytic continuation map to the case of rotating wormholes.
	stacks.iop.org /0264-9381/19/2399 (310 words)

	EN222: Mechanics of Solids - Linear Elasticity (Site not responding. Last check: 2007-10-21)
		Half-space problems can be solved using the same analytic continuation procedure that we developed for isotropic elasticity.
		This problem can be solved using exactly the same analytic continuation that we used to solve the corresponding problem for an isotropic material. The solution will be constructed using vectors of two analytic functions
		As usual, our first order of business is to find a continuation that satisfies traction and displacement continuity across the interface. Traction and displacement continuity follow as
	www.engin.brown.edu /courses/en224/anis_bvp/anis_bvp.htm (1323 words)

	SIMA Volume 35 Issue 5
		We study the families of periodic orbits of the spatial isosceles 3-body problem (for small enough values of the mass lying on the symmetry axis) coming via the analytic continuation method from periodic orbits of the circular Sitnikov problem.
		Using the first integral of the angular momentum, we reduce the dimension of the phase space of the problem by two units.
		This work is merely analytic and uses the variational equations in order to apply Poincaré's continuation method.
	epubs.siam.org /SIMA/volume-35/art_40788.html (197 words)

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