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Topic: Contour integral

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	Line integral - Wikipedia, the free encyclopedia
		The integral is then the limit of this sum, as the lengths of the subdivision intervals approach zero.
		The "path integral formulation" of quantum mechanics actually refers not to path integrals in this sense but to functional integrals, that is, integrals over a space of paths, of a function of a possible path.
		However, path integrals in the sense of this article are important in quantum mechanics; for example, complex contour integration is often used in evaluating probability amplitudes in quantum scattering theory.
	en.wikipedia.org /wiki/Line_integral (749 words)

	Examples of contour integration (Site not responding. Last check: 2007-11-01)
		In complex analysis, contour integration might be defined as integration along a curve in the complex plane, but such a definition would fail to capture much of both the flavor and the purpose of the topic.
		That is, some of the contour over which the integral is actually taken is to be disposed of later, typically by upper bounds of the integral on those stretches.
		We will evaluate it by expressing it as a limit of contour integrals along the contour C that goes along the real line from −a to a and then counterclockwise along a semicircle centered at 0 from a to −a.
	www.1-free-software.com /en/wikipedia/e/ex/examples_of_contour_integration.html (325 words)

	Contour Integral (Site not responding. Last check: 2007-11-01)
		We have a line integral, and a path integral, and now a contour integral, all running along a curve inside some kind of field.
		The contour integral of f around p is the integral of f(p(t))×p′(t).
		The contour integral is 0 whenever f is a linear function of z.
	www.mathreference.com /cx,cint.html (297 words)

	* Contour - (GIS): Definition (Site not responding. Last check: 2007-11-01)
		Contour lines connect a series of points of equal elevation and are used to illustrate topography, or relief, on a map.
		Contours are derived data, data interpolated from information of altitude at known points, and in themselves offer no information about the surface morphology between them.
		Contour interval: this is to be shown in the lower margin near the graphic scales.
	en.mimi.hu /gis/contour.html (1190 words)

	Path integral (Site not responding. Last check: 2007-11-01)
		In mathematics, a path integral (also known as a line integral) is an integral where the function to be integrated is evaluated along a path or curve.
		The integral is then the limit as the distances of the subdivision points approach zero.
		In qualitative terms, the integrand of a path integral in vector calculus can be thought of as a measure of the effect of a given vector field along a given curve.
	www.gogoglo.com /wiki/en/wikipedia/p/pa/path_integral.html (565 words)

	7.9.2 Contour integral evaluation
		Contour integrals along several different crack tips can be evaluated at any time by repeating the CONTOUR INTEGRAL* option as often as needed in the step definition.
		Contributions to the contour integral due to concentrated loads in the domain are not included; instead, the mesh must be modified to include a small element and a distributed load must be applied to this element.
		The contour integral evaluation capability in ABAQUS/Standard assumes that the elements that lie within the domain used for the calculations are quadrilaterals in two-dimensional or shell models or bricks in continuum three-dimensional models.
	www.rpi.edu /AFS/home/85/millem/public_html/v6.4/books/usb/pt03ch07s09aus47.html (4549 words)

	Contour integral
		In mathematics, a line integral (in rare cases called a path integral) is an integral where the function to be integrated is evaluated along a curve.
		The function to be integrated should be a vector-field.
		In qualitative terms, a line integral in vector calculus can be thought of as a measure of the effect of a given vector field along a given curve.
	www.algebra.com /algebra/about/history/Contour-integral.wikipedia (752 words)

	Center for Science Education:
		Our closed contour now has a curve C that begins at 3 and goes to -3, and becomes complete by virtue of the line L that runs from -3 to 3.
		Therefore, the value of (∂Q/∂x) - (∂P/∂y) is zero and thus the value of the line integral over the entire (and closed) contour C + L is zero.
		This means we know that the sum of the line integrals over C and L must sum to zero.
	www.luc.edu /faculty/dslavsk/contoursolution.shtml (135 words)

	PlanetMath: estimating theorem of contour integral (Site not responding. Last check: 2007-11-01)
		For applications of this important theorem, see the example of using residue theorem.
		"estimating theorem of contour integral" is owned by pahio.
		This is version 3 of estimating theorem of contour integral, born on 2005-06-03, modified 2005-06-22.
	www.planetmath.org /encyclopedia/EstimatingTheoremOfContourIntegral.html (119 words)

	PlanetMath: Cauchy integral theorem (Site not responding. Last check: 2007-11-01)
		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.
		The original proof makes use of this fact, and calls on Green's Theorem to conclude that the contour integral vanishes.
		This is version 10 of Cauchy integral theorem, born on 2002-08-01, modified 2005-07-09.
	planetmath.org /encyclopedia/CauchyIntegralTheorem.html (392 words)

	[No title]
		The limit must be independent of the shape of the contour C (as long as all its points approach the point P in the limit as the area a of the contour goes to zero).
		A.2.4 that the contour integral around the triangular contour in the plane perpendicular to n can also be written as the sum of three integrals around the three triangular contours in the respective coordinate planes.
		where each contour integral is denoted by the subscript taken from the unit vector normal to the plane of the contour.
	web.mit.edu /6.013_book/www/appendices/app2.html (1229 words)

	Contour Integrals
		Of course, this is based on one definition of a Riemann integral, in which the function at the left endpoint is multiplied by the length of the interval, and must be corrected at the midpoint if the interval is split in two.
		In general, the contour can be shifted across any rectangle for which a linear approximation to the function is valid, which would exclude any singularities or branch points - points where the function is not invertible and so making its definition suspect.
		The vanishing of a closed contour integral is usually verified by using Green's formula, which is a useful technique in its own right.
	delta.cs.cinvestav.mx /~mcintosh/comun/complex/node22.html (1105 words)

	sciforums.com - Cauchy's Integral Theorem
		the contour integral around any closed curve for a function that is not necessarily holomorphic inside that curve need not be zero.
		the integral of the electric field over the surface is equal to the integral of the divergence of the electric field over the volume enclosed by that surface.
		So for the contour integral of f(z) which is holomorphic (analytic) at each point in and on the contour C, the value of the integral is independent of path along the contour between the two endpoints.
	www.sciforums.com /showthread.php?t=33287 (1574 words)

	Cauchy Goursat Theorem
		The Cauchy (biography) Goursat (biography) theorem states that the contour integral of f(z) is 0 whenever p(t) is a piecewise smooth, simple closed curve, and f is analytic on and inside the curve.
		The contour integral of a constant, or z times a constant, is 0, thanks to the previous theorem, so the first two terms drop out.
		The total integral is still 0, hence the integral of f along the outer circle equals the integral of f along the inner circle.
	www.mathreference.com /cx,cgt.html (1169 words)

	[No title]
		All of this is quite standard and the results may be found in tables of integrals.
		We show this integral vanishes by setting z = e^{ix} and taking the contour integral round the unit circle C. This integral becomes (1/i) int_C log (1 + bz) dz/z.
		So the integral is the potential on the circle due to a uniform circle of diameter 1 (and density 1) in the plane, and this is equal to the potential at the centre, which is the integral from 0 to pi of ln(1/2), which is -pi ln(2).
	www.math.niu.edu /~rusin/known-math/99/contour_int (659 words)

	Resources for Teaching Complex Analysis
		Interpreting the Contour Integral This is an activity on the contour integral and the Polya Field.
		Cauchy's Theorem and Curve Deformation: In this activity students calculate the contour integral along a circle of radius two centered at the origin for a function having two poles inside this circle.
		The integral of a function f along an arc gamma is the length of the gamma multiplied by the average along gamma of f times the unit tangent vector.
	faculty.gvsu.edu /fishbacp/complex/complex.htm (1519 words)

	Path integral (Site not responding. Last check: 2007-11-01)
		This is not about "path integrals" in the sense that means that which was studied by Richard Feynman.
		In mathematics, a path integral is an integral where the function to be integrated is evaluated along a path or curve.
		The path integral is a fundamental tool in complex analysis, where it is also called a contour integral.
	www.fact-index.com /p/pa/path_integral.html (173 words)

	PlanetMath: contour integral (Site not responding. Last check: 2007-11-01)
		(iii) Reversing the direction of the curve changes the sign of the integral.
		Cross-references: piecewise smooth, continuous, rectifiable, integral, sign of, right, variable, complex, components, terms, Riemann-Stieltjes integral, Riemann integral, real axis, segment, infinity, limit, point, sum, partition, curve, image, function
		This is version 13 of contour integral, born on 2002-07-24, modified 2006-08-27.
	planetmath.org /encyclopedia/ContourIntegral.html (235 words)

	Re: contour integrals (Site not responding. Last check: 2007-11-01)
		All you need is the simple version of Cauchy's theorem, which states the contour integral of f is 0 if 1.
		f is analytic on a neighborhood of the contour and its interior.
		In this case f(z) = e^(iz)/z, which is analytic on and inside the contour you describe.
	www.talkaboutscience.com /group/sci.math/messages/892277.html (176 words)

	Analysis, Convergence, Series, Complex Analysis - Numericana
		Uniform convergence does imply that the integral of the (uniform) limit is the limit of the integrals.
		Therefore, the contribution of the outer semicircle to the contour integral tends to zero as the radius tends to infinity.
		The integral along the right part is exactly the integral we are asked to compute, whereas the left part contributes i times that quantity.
	home.att.net /~numericana/answer/analysis.htm (4031 words)

	physics - Path integral
		Important statements about path integrals are the Cauchy integral theorem and Cauchy's integral formula.
		In qualitative terms, a path integral in vector calculus can be thought of as a measure of the effect of a given vector field along a given curve.
		For example, the work done on a particle traveling on a curve C inside a force field represented as a vector field F is the path integral of F on C.
	www.physicsdaily.com /physics/Path_integral (550 words)

	existence of derivatives of all orders
		Exploiting the fact that the derivative of an integral is the integral of the derivative of its integrand,
		In fact, if f were a polynomial and the contour were a sufficiently large circle, the highest power would dominate the logarithmic derivative, provoking the conclusion that there were just as many zeroes inside, as the degree of the polynomial.
		An interesting point here is the way that the contour integral transforms the result that a power has n roots into the fact that any polynomial of that degree also has roots, although they are not necessarily (real multiples of) roots of unity.
	delta.cs.cinvestav.mx /~mcintosh/comun/complex/node24.html (378 words)

	Path integral - Wikipedia, the free encyclopedia
		Line integral, the integral of a function along a curve
		Functional integration, the integral of a functional over a space of curves
		Path integral formulation of quantum mechanics using functional integration, due to Richard Feynman
	en.wikipedia.org /wiki/Path_integral (104 words)

	ifm efector dualis - Contour verification (Site not responding. Last check: 2007-11-01)
		With a size of only 42 x 42 x 43.5 mm the efector dualis contour sensor takes us to a new dimension and can solve applications where mounting space is at a premium.
		For communication and parameter setting a USB adapter cable is used for the connection to the PC/laptop and with the operating software "efector dualis Control Panel" settings can be made and stored in the sensor.
		A new contour is set in the sensor in just 6 operating steps – easy to use.
	www.ifm-electronic.com /ifmaus/web/dualis-kontur.htm (270 words)

	nrich.maths.org::Mathematics Enrichment::NRICH
		To say what contour integration means, we first have to define path integration.
		A contour is simply a closed curve in the complex plane, and we can evaluate the contour integral as a sum of 2 path integrals.
		This integral can be evaluated by elementary means(by partial fractions) or by considering the contour integral along the semicircle with base {(r,0):-
	www.nrich.maths.org.uk /askedNRICH/edited/2169.html (349 words)

	PHY 581: Mathematical Methods for Physicists
		To evaluate the integral, we need to consider the contour integral with the contour shown below.
		This contour integral can be express in terms of four simpler contour integrals shown below.
		Finally, we compute the sum of residues of the contour integral on the LHS of the equation above.
	www.geocities.com /Paris/Shoppe/2809/PHY_581_03_01.html (501 words)

	Cauchy's Integral Formula (Site not responding. Last check: 2007-11-01)
		For simplicity assume 0 is in the interior of the contour.
		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.
		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)

	Contour Integration [Archive] - Advanced Physics Forums (Site not responding. Last check: 2007-11-01)
		We were given the value of an integral but told to have a go finding it out if we got bored....
		The contour is a sort of 3/4 circle (top right quater missing) with the origin and positive y axis cut out.
		I'm arguing that this isn't a contour integral at all.
	www.advancedphysics.org /forum/archive/index.php/t-3051.html (397 words)

	Calculation of Groundwater Integral (Site not responding. Last check: 2007-11-01)
		Series expansions are obtained for a contour integral that appears in some solutions for flow in aquifers that have a bottom impermeable boundary and are capped on top with an aquitard containing a shallow standing water table.
		These solutions describe drawdown for delayed-yield flow to a well and the stream depletion and drawdown that occur when water is pumped from a well beside a stream.
		Hunt in 2003 previously gave a series expansion for the calculation of this integral for a somewhat more limited range of its parameters.
	www.pubs.asce.org /WWWdisplay.cgi?0529094 (125 words)

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