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Topic: Maximum modulus principle

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	Springer Online Reference Works (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-09-06)
		A theorem expressing one of the basic properties of the modulus of an analytic function.
		The maximum-modulus principle is valid whenever the principle of preservation of domain is satisfied (cf.
		This principle is also called the maximum principle, cf.
	eom.springer.de.cob-web.org:8888 /M/m063110.htm (282 words)

	PlanetMath: proof of maximal modulus principle
		Taking modulus on both sides and using the estimating theorem of contour integral
		is a maximum, the last inequality must be verified by having the equality in the
		This is version 16 of proof of maximal modulus principle, born on 2006-03-15, modified 2006-09-10.
	planetmath.org /encyclopedia/ProofOfMaximalModulusPrinciple.html (258 words)

	Given n points on the unit circle...
		Maximum Modulus Principle: Let f be a nonconstant holomorphic function in the open connected subset G of C. Then absolute value of f does not attain a local maximum.
		the maximum modulus principle simply states that on a closed bound set such as the disc that the maximum occurs on the boudnary exactyl because there are no local maxima on the interior:
		THis result remains true under more general circumstances since the maximum moduls principle apples to any open connected subset of C, so as long as we are dealing with the closure of an open connected subset of C the maxmimum cannot occur on the open subset but on its boundary.
	www.physicsforums.com /showthread.php?t=83003 (984 words)

	Orðasafn: M
		maximum in the large efsta gildi, hæsta gildi, stærsta gildi, stærsta hágildi, víðfeðmt hágildi, = absolute maximum, = global maximum, = largest value, = maximal value, = maximum value, -> maximum.
		maximum in the small staðbundið hágildi, hágildi, = local maximum, = maximum 1, = relative maximum.
		maximum principle 1 (for holomorphic functions) hágildissetning, hágildislögmál, háalgildissetning, háalgildislögmál, = maximum modulus principle, = principle of maximum 1.
	www.hi.is /~mmh/ord/safn/safnM.html (2128 words)

	Maximum modulus principle - Wikipedia, the free encyclopedia
		is a local maximum for this function also, it follows from the maximum principle that
		The maximum modulus principle has many uses in complex analysis, and may be used to prove the following:
		The Fundamental theorem of algebra, as may be seen in the classic text "Introduction to Complex Analysis", by Nevanlinna and Paatero.
	en.wikipedia.org /wiki/Maximum_modulus_principle (388 words)

	PlanetMath: maximum principle
		maximal modulus principle, maximum principle for harmonic functions
		proof of weak maximum principle for real domains
		This is version 2 of maximum principle, born on 2002-06-09, modified 2004-10-23.
	planetmath.org /encyclopedia/MaximalModulusPrinciple.html (117 words)

	Springer Online Reference Works (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-09-06)
		A generalization of the maximum-modulus principle for analytic functions to the case of functions that are given a priori as unbounded; it was first given in its simplest form by E.
		The main content of the result of Phragmén and Lindelöf, in a somewhat modernized form, consists in the following two propositions, which are successive extensions of the maximum-modulus principle.
		It extends the maximum-modulus principle to functions about the behaviour of which on the boundary only partial information is available.
	eom.springer.de.cob-web.org:8888 /p/p072630.htm (518 words)

	The maximum modulus principle (Site not responding. Last check: 2007-09-06)
		This series of inequalities relates the value of the function (as well as its derivatives) at the center of a circle of analyticity to the maximum value on the circumference of the circle.
		Enclosing it within a circle small enough to still lie in the domain, we observe that there must be a point on the circumference of that little circle with a still greater modulus, which would be a contradiction, or all the values on that little circle would equal the purportedly greater modulus.
		The final result of all this reasoning is the maximum modulus principle, that the maximum absolute value of an analytic function lies on the boundary of any domain of analyticity.
	delta.cs.cinvestav.mx /~mcintosh/comun/complex/node26.html (549 words)

	PlanetMath: proof of maximal modulus principle (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-09-06)
		PlanetMath: proof of maximal modulus principle (via CobWeb/3.1 planetlab2.isi.jhu.edu)
		"proof of maximal modulus principle" is owned by cvalente.
		Cross-references: constant function, formula, point, calculate, domain, entire, curve, circumference, constant, implies, equality, inequality, estimating theorem of contour integral, sides, modulus, canonical, Cauchy integral formula, boundary, interior, compact, continuous, holomorphic
	planetmath.org.cob-web.org:8888 /encyclopedia/ProofOfMaximalModulusPrinciple.html (267 words)

	Springer Online Reference Works (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-09-06)
		The basic property here is that of openness of the image, which follows from Rouché's theorem or from the principle of the argument (cf.
		The principle of preservation of domain can be considered as a generalization of the maximum-modulus principle for holomorphic functions.
		The principle of preservation of domain holds for holomorphic functions on an arbitrary complex manifold: the set of values of any non-constant holomorphic function on a connected complex manifold
	eom.springer.de.cob-web.org:8888 /P/p074440.htm (288 words)

	Complex Analysis 3 at the University of Zimbabwe (Site not responding. Last check: 2007-09-06)
		The course continues with full presentation of some theorems, considered in the previous course without proofs --- Maximum Modulus Principle, Logarithmic Indicator, Rouche's Theorem.
		M\"obius transforms on the extended complex plane - definition, particular cases, properties; reflection (inversion) with respect to "generalized circumferences" (circumferences and stright lines) - properties, modulus of a doubly connected domain; classification of the M\"obius transforms.
		Maximum modulus principle, Schwarz's lemma, Logarithmic indicator, Argument principle, Rouche's theorem.
	www.uz.ac.zw /science/maths/courses/hmth322.htm (255 words)

	Maximum principle - Wikipedia, the free encyclopedia
		In mathematics, the maximum principle in harmonic analysis states that if f is a harmonic function, then f cannot exhibit a true local maximum within the domain of definition of f.
		By replacing "maximum" with "minimum" and "larger" with "smaller", one obtains the minimum principle for harmonic functions.
		The maximum principle also holds for the more general subharmonic functions, while superharmonic functions satisfy the minimum principle.
	en.wikipedia.org /wiki/Maximum_principle (280 words)

	Math 403:02 diary, spring 2005
		Maximum modulus principle (version 2) If f(z) is an analytic function defined on a bounded domain together with the boundary of the bounded domain, and if f(z) is continuous on this set (the domain together with its boundary) then the maximum of f(z)
		Any function which is continuous on z&;t;=1 and analytic on z<1 attains its maximum in modulus on the boundary.
		The boundary (the imaginary axis) is mapped to the unit circle, and there the modulus is 1.
	www.math.rutgers.edu /~greenfie/mill_courses/math403/diary2.html (10921 words)

	hill pubs
		A sharp maximum principle for degenerate elliptic-parabolic equations, Indiana Univ. Math.
		The maximum modulus principle, I. Necessary Conditions (with D. Ellis and C. Seabury), Indiana Univ. Math.
		The maximum modulus principle for CR functions on smooth real embedded submanifolds on C^n (with D. Ellis), Proc.
	www.math.sunysb.edu /~Dhill/pubs.html (1109 words)

	Maximum number of expressions in (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-09-06)
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	maximum-number-of-expressions-in.dirkdebruyne.be.cob-web.org:8888 (169 words)

	La taille maximum de la mémoire partagée (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-09-06)
		la taille maximum de la mémoire partagée of the countrie.
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	la-taille-maximum-de-la-memoire-partagee.dirkdebruyne.be.cob-web.org:8888 (156 words)

	[No title]
		Berhanu: Liouville's theorem and the maximum modulus principle for a system of complex vector fields.
		Berhanu: Extreme points and the strong maximum principle for CR functions, Contemporary Math., Volume 205, 1997, 1-13.
		S. Berhanu and C. Wang: On the maximum principle and a notion of plurisubharmonicity for abstract CR manifolds, Michigan Mathematical Journal, to appear
	www.math.temple.edu /~berhanu/recpub.html (389 words)

	Complex Modulus -- from Wolfram MathWorld (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-09-06)
		The complex modulus is implemented in Mathematica as
		Complex Number, Imaginary Part, Maximum Modulus Principle, Minimum Modulus Principle, Real Part.
		Krantz, S. "Modulus of a Complex Number." §1.1.4 n Handbook of Complex Variables.
	mathworld.wolfram.com.cob-web.org:8888 /ComplexModulus.html (115 words)

	Maximum length of command line arguments (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-09-06)
		Maximum length of command line arguments (via CobWeb/3.1 planetlab2.isi.jhu.edu)
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	maximum-length-of-command-line-arguments.dirkdebruyne.be.cob-web.org:8888 (83 words)

	Applied Mathematics Comprehensive, Part 1
		Line integrals, Cauchy’s theorem, Cauchy’s integral formula, power series representation and consequences, uniqueness theorem, mean value theorem, maximum modulus principle, open mapping theorem.
		Wave equation in 3D: spherical means, Huygen's principle, energy method, Duhamel's formula.
		Laplace and Poisson equations: mean-value theorem, maximum principle, Green's identities and applications, Dirichlet and Neumann problems.
	www.math.mcmaster.ca /~boden/Grad/AppliedMath1.html (399 words)

	PhD Requirements
		Complex integration, Cauchy integral theorem and formulas, Morera's theorem, Liouville's Theorem, maximum modulus principle.
		Residue theorem, evaluation of definite integrals, argument principle.
		Dirichlet and Neumann problems for Laplace, Helmholtz and heat equations, maximum principle and uniqueness theorems for elliptic and parabolic equations
	www.math.buffalo.edu /gr_reqts_phd.html (1504 words)

	Message maximum (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-09-06)
		maximum variable size allowed by the program is exceeded
		to start a managed process after the maximum retry
		She was of his fields, as each one flower is as physical phenomena to be discussed (without waiting for others,) program causes a smile - and so does the her.
	message-maximum.dirkdebruyne.be.cob-web.org:8888 (73 words)

	Complex.html
		zeros and singularities of analytic functions, maximum modulus
		principle, conformal mapping, Schwarz's lemma, the residue theorem,
		× Inverse mapping theorem, open mapping theorem, local maximum modulus principle.
	www.math.rutgers.edu /~costin/503 (159 words)

	Theorems of Morera and Liouville and Extensions
		We now prove an important result concerning the modulus of an analytic function.
		We sometimes state the maximum modulus principle in the following form.
		assumes its maximum value, and does so only at point(s)
	math.fullerton.edu /mathews/c2003/LiouvilleMoreraGaussMod.html (251 words)

	Preliminary Examinations And Basic Graduate Sequences
		Measure theory and integration: Lebesgue measure theory, abstract measure theory, measurable functions, integration, space of absolutely integrable functions, dominated convergence theorem, convergence in measure, Riesz representation theorem, product measure the Fubini theorem, Lpspaces, derivative of a measure and Radon-Nikodym theorem, fundamental theorem of calculus,.
		Review of the basic theory of one complex variable, the Cauchy-Riemann equations, Cauchy's theorem, power series expansions, the maximum modulus principle, Classification of singularities, Residue theorem, argument principle, harmonic functions, linear fractional transformations, Conformal mappings, Riemann mapping theorem, Picard's theorem, introduction to Riemann surfaces.
		Principle bundles, associated bundles and vector bundles, connections on principle and vector bundles.
	www.math.ucsc.edu /graduate/preexam_seq.html (1021 words)

	Art of Problem Solving Forum
		Well I found that such a function has at least one zero (this is rather trivial via the maximum modulus principle), but I still cannot see (even intuitively) where the second zero comes from.
		Also, that should be "counting multiplicity"- I can construct an example that has one zero of multiplicity two.
		By the argument principle, the number of zeros of
	www.artofproblemsolving.com /Forum/post-402845.html (462 words)

	San Francisco State University Department of Mathematics Course Syllabus MATH 380 Introduction to Functions of a ... (Site not responding. Last check: 2007-09-06)
		Students who successfully complete this course should be capable of:
		Applying the Cauchy-Riemann Equations, Cauchy’s Theorem, Cauchy’s Integral Formula, Cauchy’s Inequality, Liouville’s Theorem and the Maximum Modulus Principle to complex valued functions.
		Contour Integrals of Complex Functions: Cauchy’ Theorem, Cauchy’s Integral Formula, Cauchy’s Inequalities, Liouville’ Theorem, Morera’s Theorem, Maximum Modulus Theorem and elementary properties of harmonic functions
	math.sfsu.edu /stat/F03math380.htm (342 words)

	[No title]
		Topics: Mathematical Models & Origins of PDE Integral Cons.
		Mean Value Property Subharmonic and Superharmonic Extended Maximum Principles (2) Uniqueness Results -- Laplace Equation Dirichlet; Neumann; Exterior A Priori Extimates -- Laplace Equation Maximum Principle for Diffusion Equation Uniqueness Results -- Diffusion Equation Dirichlet; Neumann; Exterior A Priori Estimates -- Diffusion Equation D'Alembert Solution -- Second Order Wave Eq.
		Remainder for testing and discussion.) M546 Partial Differential Equations II Credits: 3 (3-0-0) Term Offered: Spring Description: This course focuses mainly on analytical techniques for solving and analyzing partial differential equations.
	www.colostate.edu /Depts/Mathematics/500levelClasses (2015 words)

	MATH514 - Analysis I
		One semester will be devoted to real analysis, covering such topics as Lebesgue measure and integration on the line, abstract measure spaces and integrals, product measures, decomposition and differentiation of measures, and elementary functional analysis.
		One semester will be devoted to complex analysis, covering such topics as analytic functions, power series, Mobius transformations, Cauchy's integral theorem and formula in its general form, classification of singularities, residues, argument principle, maximum modulus principle, Schwarz' lemma, and the Riemann mapping theorem.
		Level: GRAD Credit: 1 Gen Ed Area Dept: NONE Grading Mode: Graded
	www.wesleyan.edu /wesmaps/course0304/math514s.htm (155 words)

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