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Topic: Cauchy integral formula

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Complex analysis

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Richard Feynman

	PlanetMath: Cauchy integral formula
		The following technical generalization of the formula is needed for the treatment of removable singularities.
		The two concepts are, in fact, equivalent, but the standard proof of this fact uses the Cauchy Integral Formula with the (apparently) weaker holomorphicity hypothesis.
		This is version 21 of Cauchy integral formula, born on 2001-12-28, modified 2004-03-15.
	planetmath.org /encyclopedia/CauchyIntegralFormula.html (255 words)

	PlanetMath: Cauchy integral theorem
		Cauchy's theorem is an essential stepping stone in the theory of complex analysis.
		It is required for the proof of the Cauchy integral formula, which in turn is required for the proof that the existence of a complex derivative implies a power series representation.
		This is version 10 of Cauchy integral theorem, born on 2002-08-01, modified 2005-07-09.
	planetmath.org /encyclopedia/CauchyIntegralTheorem.html (393 words)

	CAUCHY, A.L.(1789-1857)
		Cauchy was born in Paris in 1789 and recceived his early education from his father.
		Cauchy wrote extensively and profoundly in both pure and applied mathe matics, and he can probably be ranked next to Euler in volume of output.
		Cauchy's work exhibits great attention to rigor, and as such was largely reaponsible for inspiring other mathematicians to attempt the banishment of blind formal manipulation and of intuitive proofs from analysis.
	library.thinkquest.org /22584/temh3041.htm (493 words)

	Augustin-Louis Cauchy
		Modern mathematics is indebted to Cauchy for two of its major interests, each of which marks a sharp break with the mathematics of the eighteenth century.
		Cauchy was the oldest of six children of a Catholic lawyer, classical scholar, police officer and supporter of the king.
		Cauchy was very pious all of his life -- a trait which many of his contemporaries thought he overdid.
	scidiv.bcc.ctc.edu /Math/Cauchy.html (861 words)

	Cauchy's integral theorem - Wikipedia, the free encyclopedia
		In mathematics, the Cauchy integral theorem in complex analysis, named after Augustin Louis Cauchy, is an important statement about line integrals for holomorphic functions in the complex plane.
		This is significant, because one can then prove Cauchy's integral formula for these functions, and from that one can deduce that these functions are in fact infinitely often continuously differentiable.
		The Cauchy integral theorem is considerably generalized by the Cauchy integral formula and the residue theorem.
	en.wikipedia.org /wiki/Cauchy_integral_theorem (457 words)

	Springer Online Reference Works
		One definition, which was originally proposed by Cauchy, and was considerably advanced by Riemann, is based on a structural property of the function — the existence of a derivative with respect to the complex variable, i.e.
		Cauchy's integral theorem yields Cauchy's integral formula, which expresses the values of an analytic function inside a domain in terms of its values on the boundary:
		An ample supply of integral formulas, including formulas with an analytic kernel for many domains, is contained in the general Leray formula.
	eom.springer.de /A/a012240.htm (4938 words)

	Cauchy's integral formula - Wikipedia, the free encyclopedia
		Moreover, just as in the case of the Cauchy integral theorem, it is sufficient to require that f be holomorphic in the open region enclosed by the path and continuous on that region's closure.
		These formulas can be used to prove the residue theorem, which is a far-reaching generalization.
		By using the Cauchy integral theorem, one can show that the integral over C (or the closed rectifiable curve) is equal to the same integral taken over a tiny circle around a.
	en.wikipedia.org /wiki/Cauchy's_integral_formula (562 words)

	MA231 Vector Analysis
		Cauchy's theorem for complex differentiable functions is then established by means of the main integral theorems of vector calculus.
		Cauchy's integral formula which expresses the value of a complex differentiable function at a point as a line integral of the function on a contour surrounding the point is the key result from which the stunning properties of complex differentiable functions follow.
		Be able to prove Cauchy's integral formula from Cauchy's theorem, and to use the integral formula to establish differentiability and series properties of complex differentiable functions;
	www.maths.warwick.ac.uk /pydc/green/green-MA231.html (632 words)

	Cauchy's Integral Theorem (Site not responding. Last check: 2007-11-03)
		Cauchy's formula says that values of f and its derivatives interior to C are completely determined by values of f on C.
		This formula can be used to easily evaluate certain integrals along simple closed curves.
		These integrals arise in many applications, such as flow of electricity or fluids in two dimensions.
	www.math.unm.edu /cauchy/cauchy_desc.html (95 words)

	Integral Representations for Analytic Functions
		In Chapter 7, we use the Cauchy integral formulas to prove
		The Cauchy integral formulas are a convenient tool for evaluating certain contour integrals.
		We manipulate the integral and use Cauchy's integral formula to obtain
	math.fullerton.edu /mathews/c2003/IntegralRepresentationMod.html (235 words)

	Complex Variables
		Using the formulas for the trig functions of the sum and difference of two angles, it is easy to prove that (cos θ + i sin θ)(cos θ' + i sin θ') = cos(θ + θ') = i sin(θ + θ').
		The elegant fundamental theorem of Cauchy's theory is ∫ f(z)dz = 0, where the integral is taken about a closed curve C surrounding a region where f'(z) exists, as shown in the diagram.
		The next step is to obtain an integral formula for f(z) and its derivatives, where z is any point in a region where f(z) is analytic, and the integration is carried out on the boundary of the region.
	www.du.edu /~jcalvert/math/complex.htm (4434 words)

	Complex Variables
		Suppose the complex function f(z) is differentiable on and within a closed curve C in the complex plane, and let R be the region inside the closed curve C.
		This is the same as the value of the integral along the straight path from 1 to i.
		The two values of the integral are not equal because 1/z is not differentiable at z = 0, and the point 0 is in the region enclosed by the two paths.
	www.jgsee.kmutt.ac.th /exell/Numbers/CplxVar.htm (998 words)

	Graduate Math Courses
		Consequences of Cauchy's theorem: Cauchy's integral formula, Liouville's theorem, fundamental theorem of algebra, Cauchy's formula for derivatives and Morera's theorem.
		Complex differentiation, Cauchy-Riemann equations, Cauchy integral formula, Taylor and Laurent expansions, residue theory, contour integration including branch point contours, uses of Jordan's lemma, Fourier and Laplace transform integrals, conformal mapping.
		Problems studied in this way include the most naturally formulated as integral equations over relatively high dimensional phase spaces, as well as those in which estimates of integrals of functions of a large number of variables are sought.
	www.cgu.edu /print/628.asp (2740 words)

	The Complex Variable Boundary Element Method (More)
		The Complex Variable Boundary Element Method or CVBEM is a generalization of the Cauchy integral formula into a boundary integral equation method or BIEM.
		In a Symposium on the Application and Numerical Solution of Integral Equations (1978), two papers were presented which address the use of the Cauchy singular integral equation for the solution of potential problems.
		In another application, the time derivative was also included in the work of Brevig et al (1982) where a Cauchy integral model is used to approximate the time evolution of two-dimensional seawater waves and associated forces.
	www.hromadka.net /COMPLE1_More.html (869 words)

	Cauchy Integral Formula and Electrodynamics
		Is it possible to solve for an E field from a charge density function using the Cauchy Integral Formulas from complex variables?
		If I remember that was a curl type formula and you need a divergence type formula.
		I was thinking that just like the Gauss's theorem (the surface integral version of the Div[E] = rho/ epsilon) picks out charges which are in effect mathematical singularities, so to the cauchy residue theorem picks out every 1/z of a function.
	www.physicsforums.com /showthread.php?t=43964 (173 words)

	Cauchy Integral Formula (Site not responding. Last check: 2007-11-03)
		Note: Analyticity is sufficient for this integral to be zero, but not necessary.
		One of the strongest properties of the Cauchy Integral Theorem is that it allows us to deform the contour C. As long as C remains within D, the value of the integral is unchanged because the function is path independent on that domain.
		The Cauchy Integral Formula says that for a function f(z) which is analytic in a simply connected domain D, on a simple closed path C in D where z0 lies in D, then the following equations holds/TD>
	web.mit.edu /bnbond/www/MathSolverChart/IntCauchy.html (165 words)

	YORK UNIVERSITY
		State Cauchy's Integral Theorem and Cauchy's Integral Formula and say why they are important in complex analysis.
		lies inside C. Cauchy's Integral Theorem shows that under appropriate conditions an integral of a function is independent of path.
		This makes possible the idea of an indefinite integral and leads to the possibility of deforming a contour and Cauchy's Integral Formula which shows, among other things that an analytic function is determined by its values in a relatively small set of points and that an analytic function is infinitely differentiable.
	www.math.yorku.ca /Who/Faculty/Muldoon/cv/cv99tr.htm (504 words)

	[No title]
		I concluded that the integral of dz/z along any positively oriented simple closed loop around the origin is 2pi i.
		Principle of deformation of paths: integrals of a holomorphic function does not depend on the path if the function is holomorphic IN BETWEEN the two paths.
		The residue theorem implies the Cauchy's integral formula.
	math.stanford.edu /~oprea/106.html (1168 words)

	Math 413 Homework
		Study Cauchy’s generalized integral formula (Theorem 19) and Examples 4 and 5 on pp.
		Study the Cauchy estimates for the derivatives of an analytic function (Theorem 20) and Liouville’s Theorem (Theorem 21) on pp.
		317: 1 and 2 (but we are using Cauchy’s integral formula instead of Cauchy’s residue theorem).
	www.cbu.edu /~lbecker/Hwk/413_HwkF06.htm (587 words)

	EN224: LINEAR ELASTICITY (Site not responding. Last check: 2007-11-03)
		This procedure actually gives exact solutions for any displacement or traction boundary value problem, although the resulting integrals can be hard to evaluate unless the solid can be mapped conveniently to the unit circle.
		To evaluate the second term, expand the complex potential as a Taylor series (we know this can be done, since the potentials must be analytic within the disk).
		The first integral on the right hand side may be evaluated directly using the Cauchy integral formula.
	www.engin.brown.edu /courses/en224/complexchy/complexchy.html (547 words)

	Mathematics Other Homework Help
		flux integral - Let {see attachment} and let n be the outward unit normal vector to the positively oriented circle {see attachment}.
		Line integral - a) For what simple closed(positively oriented) curve C in the plane does the line integral of (e^(-x)+ 4x^2y +y)dx + (x^3-x*y^2+5x)dy has the largest positive value?
		Cauchy Integral Formula - (See attached file for full problem description and imbedded formulae) --- Why can (1) be regarded as a special case of (2)?
	www.brainmass.com /homeworkhelp/math/other/38759 (209 words)

	Amazon.com: Complex Analysis: Books: Joseph Bak,Donald J. Newman (Site not responding. Last check: 2007-11-03)
		I mean, he pays no attention to the most recent and elegant refinements of the basic theory, so the student is not immediately able to understand the real important ideas behind the subject.
		For example, nowadays the proof of the Cauchy integral formula is presented as a more ar less easy corollary of the general Stokes theorem.
		The Cauchy integral theorem is also obtained easily following the same fashion.
	www.amazon.com /Complex-Analysis-Joseph-Bak/dp/0387947566 (1856 words)

	Table of contents for Library of Congress control number 2001046415
		The Cauchy Integral Formula and the Cauchy Integral Theorem 43 §2.5.
		An Introduction to the Cauchy Integral Theorem and the Cauchy Integral Formula for More General Curves 53 Exercises 60 Chapter 3.
		The Cauchy Estimates and Liouville's Theorem 85 §3.5.
	www.loc.gov /catdir/toc/fy031/2001046415.html (548 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		] The Cauchy data for a hyperbolic partial differential equation consist of the value of the field and its time derivative on some spacelike surface.
		] A semiempirical formula for the index of refraction n of a medium as a function of wavelength
		] The theorem expressing a line integral around a closed curve of a function which is analytic in a simply connected domain containing the curve, except at a finite number of poles interior to the curve, as a sum of residues of the function at these poles.
	www.accessscience.com /Dictionary/C/C12/DictC12.html (1783 words)

	Cauchy's Integral Formula (Site not responding. Last check: 2007-11-03)
		Since f(z)/z is analytic between these two closed curves, apply the Cauchy Goursat theorem, and the integral of f around our original closed curve is equal to the integral of f around the circle of radius r.
		Remember that f is continuous at 0, so for small r, f is arbitrarily close to w.
		Therefore the integral around the tiny circle of radius r, and the integral around our original path, is f(0)×2πi.
	www.mathreference.com /cx,cif.html (186 words)

	Connexions - Search Repository (Site not responding. Last check: 2007-11-03)
		Approximation Formulae for the Gaussian Error Integral, Q(x)
		This module gives a defintion of Cauchy's Integral Formula and describes its use and usefulness.
		Introduction to a fundatmental principle of integral calculus- the area of a function.
	cnx.org /content/search?words=integral (224 words)

	cauchy - OneLook Dictionary Search
		We found 3 dictionaries with English definitions that include the word cauchy:
		Tip: Click on the first link on a line below to go directly to a page where "cauchy" is defined.
		Phrases that include cauchy: cauchy sequence, cauchy integral formula, cauchy integral theorem, cauchy product, cauchy schwarz inequality, more...
	www.onelook.com /?w=cauchy (88 words)

	Complex-Analysis
		Contour integration, for example, provides a method of computing difficult integrals by investigating
		Cauchy's Integral Formula, uniform convergence of analytic functions, and the winding number
		Elementary extension phenomena: Extensions by Cauchy's integral formula, and Laurent series
	www.adwan.net /Complex-Analysis.html (480 words)

	EN224: LINEAR ELASTICITY (Site not responding. Last check: 2007-11-03)
		5.4 Solving Anti-Plane Shear Problems using Cauchy Integral Formulae
		The Cauchy integral formula sometimes allows us to find explicit solutions to Laplace’s equation.
		For example, consider a displacement boundary value problem for a circular cylinder in anti-plane shear.
	www.engin.brown.edu /courses/en224/antipcauchy/antipcauchy.html (71 words)

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