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# Topic: Cauchy principal value

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 Cauchy principal value - Wikipedia, the free encyclopedia In mathematics, the Cauchy principal value of certain improper integrals is defined as either The former is the Cauchy principal value of the otherwise ill-defined expression The former is the principal value of the otherwise ill-defined expression en.wikipedia.org /wiki/Cauchy_principal_value   (184 words)

 Cauchy, Augustin-Louis - Hutchinson encyclopedia article about Cauchy, Augustin-Louis   (Site not responding. Last check: 2007-11-06) Cauchy was born in Paris, studied engineering there and worked for a time in construction, then became a professor at the Ecole Polytechnique in 1816 and later at the Collège de France. From 1848 to 1852 Cauchy was a professor at the Sorbonne. In 1805 Cauchy provided a simple solution to the problem of Apollonius, namely to describe a circle touching three given circles, and in 1816 he published a paper on wave modulation. encyclopedia.farlex.com /Cauchy%2c+Augustin-Louis   (374 words)

 Cauchy integral theorem   (Site not responding. Last check: 2007-11-06) The Cauchy integral theorem in complex analysis is an important statement about path integral s for holomorphic function s in the complex plane. The Cauchy integral theorem is considerably generalized by the residue theorem. Cauchy - Augustin-Louis Cauchy (1789-1857) (Catholic Encyclopedia) Theory of polyhedra, symmetrical functions, proof of a theorem of Fermat which had baffled mathematicians like Gauss and Euler. www.serebella.com /encyclopedia/article-Cauchy_integral_theorem.html   (672 words)

 PlanetMath: Cauchy principal part integral   (Site not responding. Last check: 2007-11-06) In fact, this viewpoint can be used to somewhat vastly increase the set of functions for which this principal value is well-defined, such as functions that are integrable, satisfy a Lipschitz condition at 0, and whose behavior for large "Cauchy principal part integral" is owned by mathcam. This is version 7 of Cauchy principal part integral, born on 2003-07-18, modified 2005-01-25. planetmath.org /encyclopedia/CauchyPrinciplePartIntegral.html   (234 words)

 Cauchy distribution - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-11-06) As a probability distribution, it is known as the Cauchy distribution while among physicists it is known as the Lorentz distribution or the Breit-Wigner distribution. The Cauchy distribution is often cited as an example of a distribution which has no mean, variance or higher moments defined, although its mode and median are well defined and are both equal to x The Cauchy distribution is an infinitely divisible probability distribution. www.americancanyon.us /project/wikipedia/index.php/Cauchy_distribution   (698 words)

 Integral - Wikipédia Therefore the value of the integral is F(b)-F(a). Improper integrals usually turn up when the range of the function is infinite or, in the case of the Riemann integral, when the domain is infinite. One common example of an improper integral is the Cauchy principal value. su.wikipedia.org /wiki/Integral   (1270 words)

 Encyclopedia: Cauchy distribution   (Site not responding. Last check: 2007-11-06) In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. If a probability distribution has a density function f(x) then the mean or expected value is In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. www.nationmaster.com /encyclopedia/Cauchy-distribution   (1932 words)

 Encyclopedia: Principal value   (Site not responding. Last check: 2007-11-06) In considering complex multiple-valued functions in complex analysis, the principal values of a function are the values along one chosen branch of that function, so it is single-valued. For simplicity, the branch corresponding to k=0 is known as the principal branch, and along this branch, the values the function takes are known as the principal values. Determining the principal value of some of these functions can be obtained by decomposing the function into simpler ones whereby the principal value of the simple functions are straightforward to obtain. www.nationmaster.com /encyclopedia/Principal-value   (436 words)

 Improper integral - Wikipedia, the free encyclopedia In some cases, the integral from a to c is not even defined, because the integrals of the positive and negative parts of f(x) dx from a to c are both infinite, but nonetheless the limit may exist. Such cases are "properly improper" integrals, i.e., their values cannot be defined except as such limits. From the point of view of mathematical analysis it is not necessary to interpret it that way, since it may be interpreted instead as a Lebesgue integral over the set (0, ∞). www.wikipedia.org /wiki/Improper_Riemann_integral   (561 words)

 Cauchy distribution -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-06) The Cauchy distribution is an (Click link for more info and facts about infinitely divisible) infinitely divisible probability distribution. Various results in probability theory about expected values, such as the strong ((statistics) law stating that a large number of items taken at random from a population will (on the average) have the population statistics) law of large numbers, will not work in such cases. Without a defined mean, it is impossible to consider the (The second moment around the mean; the expected value of the square of the deviations of a random variable from its mean value) variance or (The square root of the variance) standard deviation of a standard Cauchy distribution. www.absoluteastronomy.com /encyclopedia/C/Ca/Cauchy_distribution.htm   (774 words)

 [No title] - k.u.leuven c***purpose the routine calculates an approximation result to a c cauchy principal value i = integral of f*w over (a,b) c (w(x) = 1/((x-c), c.ne.a, c.ne.b), hopefully satisfying c following claim for accuracy c abs(i-result).le.max(epsabe,epsrel*abs(i)). c***description c c computation of a cauchy principal value c standard fortran subroutine c real version c c c parameters c on entry c f - real c function subprogram defining the integrand c function f(x). one can allow more sub- c divisions by increasing the value of limit c (and taking the according dimension c adjustments into account). www.netlib.org /quadpack/qawc.f   (695 words)

 Cauchy distribution - Encyclopedia.WorldSearch   (Site not responding. Last check: 2007-11-06) The Cauchy distribution is the Student's t-distribution with just one degree of freedom. If (1) is construed as an improper integral rather than a Lebesgue integral, then (2) is undefined, and (1) is not necessarily well-defined. :and this is its Cauchy principal value, and it is zero, but we could also take (1) to mean, for example, encyclopedia.worldsearch.com /cauchy_distribution.htm   (687 words)

 Cauchy principal value -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-06) In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, the Cauchy principal value of certain (Click link for more info and facts about improper integral) improper integrals is defined as either The Cauchy principal value of a function can take on several nomenclatures, varying for different authors. These pathologies do not afflict (Click link for more info and facts about Lebesgue-integrable) Lebesgue-integrable functions, that is, functions the integrals of whose (A numerical value regardless of its sign) absolute values are finite. www.absoluteastronomy.com /encyclopedia/c/ca/cauchy_principal_value.htm   (201 words)

 cauchy principal value The Cauchy principal value of a finite integral of a function f about a point c with is given by. In mathematics, the Cauchy principal value of certain improper integrals is defined as either the finite numberandlt;mathandgt;\lim_{\ varepsilon\rightarrow 0+} \left … On quadrature for Cauchy principal value integrals of oscillatory … www.marymags.net /cauchy-principal-value.html   (406 words)

 [No title]   (Site not responding. Last check: 2007-11-06) Examples include the Cauchy dist. in statistics, which has a mean in the CPV sense but not in the classical sense, and examples in PDEs (say, via complex maps) where this is a physically meaningful interpretation. The usual sense of the doubly improper integral is a more restrictive notion--notice that the CPV can give a finite value to the integral of a function which is unbounded as x-->+/- infinity, which may seem a bit strange. Yes, and what's worse, the CPV is _not_ a "stable" value for the integral; if you evaluate the integral using change-of-variables you might get a different answer. math.niu.edu /~rusin/known-math/99/principal_val   (392 words)

 Maybe this Explains the Economic Cycle... best Cauchy Principal Value   (Site not responding. Last check: 2007-11-06) Cauchy's principal value of local times of Lévy processes with no negative jumps via continuous branching processes Cauchy's principal value of local times of Lévy processes with no negative jumps via continuous branching processes this paper... In mathematics, the Cauchy principal value of certain improper integrals is defined as either the finite number where b... Cauchy Principal Value -- from MathWorld Cauchy Principal Value -- from MathWorld The Cauchy principal value of a finite integral of a function f about a point c with a\leq c\leq b is given by PV \int_a^b f(x)\,dx \equiv\lim_{\epsilon\to 0... ascot.pl /th/Fourier3/Cauchy-Principal-Value.htm   (596 words)

 Cauchy's integral formula A Cauchy's integral formula for functions with values in a universal Clifford algebra and its applications. Cauchy formula and Cauchy-type integral for a class of generalized analytic functions. Cauchy's integral formula in the theory of analytic functions of several complex variables. mathews.ecs.fullerton.edu /c2003/CauchyIntegraBib/Links/CauchyIntegraBib_lnk_3.html   (588 words)

 Cauchy distribution The Cauchy distribution is a probability distribution with probability density function The Cauchy distribution is often cited as an example of a distribution which has no mean, variance or higher moments defined, although its mode and median are well defined and both zero. This example serves to show that the hypothesis of finite variance in the central limit theorem cannot be dropped (although it can be replaced with other, in some cases weaker, assumptions). en.mcfly.org /Cauchy_distribution   (459 words)

 Math Forum - Ask Dr. Math   (Site not responding. Last check: 2007-11-06) In both cases, the idea is to consider the improper integrals as limits that are symmetrically computed about the point of discontinuity (in the first case, the point of discontinuity is at infinity). In light of this, you are probably wondering why the improper integral itself is not defined to be the Cauchy principal value. Still, the CPV exists and is not without its uses, for example, in the computation of Hilbert transforms, or other more advanced topics in real and complex analysis. www.mathforum.org /library/drmath/view/61246.html   (1045 words)

 Cauchy Distribution Encyclopedia Article, Definition, History, Biography   (Site not responding. Last check: 2007-11-06) It is also an example of a more generalized version of the central limit theorem that is characteristic of all Lévy skew alpha-stable distributions, of which the Cauchy distribution is a special case. The location-scale family to which the Cauchy distribution belongs is closed under linear fractional transformations with real coefficients. In this connection, see also McCullagh's parametrization of the Cauchy distributions. www.artquilt.com /encyclopedia/Cauchy_distribution   (832 words)

 Talk:Elliptic function - Wikipedia, the free encyclopedia As written, this series is improper and divergent; but it can be made convergent by taking the Cauchy principal value, which is the limit as x->∞ of the sum of those terms with Degenerate elliptic functions and curves are obtained by setting a or b to infinity. If a or b is infinite, but not both, the Cauchy principal value diverges and other means must be used to define the function. en.wikipedia.org /wiki/Talk:Elliptic_function   (464 words)

 BioMedical Engineering OnLine | Full text | Multimodal pressure-flow method to assess dynamics of cerebral ... The remaining lower frequency BP signal, termed the "residual BP" or BP (shown as thick curve in Figure 1A), was used to identify the maximum and minimum values during the VM. values and the phases for the left MCA in the normotensive and hypertensive group, and the stroke side MCA in the stroke group. Average BP values were higher in the stroke and hypertensive groups compared to the normotensive group at baseline (p = 0.001), at BP minimum (p = 0.054) and at BP maximum (p = 0.0001) (Figure 3C). www.biomedical-engineering-online.com /content/3/1/39   (5242 words)

 4036guide Cauchy's Theorem on an arbitrary open set O, stated and proved in Section 3.2, is quite powerful and easy to use. In this version of Cauchy's Theorem, the condition that O be convex is replaced by the condition that the contour have winding number zero with respect to every point outside O ("the contour is homologous to zero in O"). Pay special attention to the boundary value problems in which the domain is the upper half-plane, so that the boundary is the real axis; and in which the boundary-value function takes on a finite number of values, each on an interval of the real axis. www.math.lsu.edu /~mcgehee/4036/guide.html   (3660 words)

 Papers on quadrature   (Site not responding. Last check: 2007-11-06) G.H. Behforooz, The use of spline-on-spline for the approximation of Cauchy principal value integrals, Appl. Diethelm, Peano kernels and bounds for the error constants of Gaussian and related quadrature rules for Cauchy principal value integrals, Numer. Dagnino and P. Lamberti, Numerical evaluation of Cauchy principal integrals based on local spline approximation operators, J. Comput. www.cs.kuleuven.ac.be /~ronald/Publi/PUB96/node4.html   (293 words)

 [No title]   (Site not responding. Last check: 2007-11-06) The CPV is the limit (as B-->inf.) of the integral from -B to B of f(x). Hence, the CPV of the doubly improper integral of f(x)=x exists because the integral from -B to B of f(x)=x is always zero (the neg. wrote: [ valid answer deleted; it carried the reminder that ] >The CPV is the limit (as B-->inf.) of the integral from -B to B of f(x). www.math.niu.edu /~rusin/known-math/99/principal_val   (392 words)

 MATH3401 Complex Analysis   (Site not responding. Last check: 2007-11-06) Since the degree of the numerator is 2 less than the degree of the denominator, the value of the integral over C Thus the Cauchy principle value of the improper integral is: provided the Cauchy principal value of the integral on the right-hand side exists, since the integrand on the left is even. www.maths.uq.edu.au /courses/MATH3401/MATH3401_WorkedSolns11.htm   (705 words)

 Cauchy Principal Value Encyclopedia Article, Definition, History, Biography   (Site not responding. Last check: 2007-11-06) Looking For cauchy principal value - Find cauchy principal value and more at Lycos Search. Find cauchy principal value - Your relevant result is a click away! Look for cauchy principal value - Find cauchy principal value at one of the best sites the Internet has to offer! www.karr.net /search/encyclopedia/Cauchy_principal_value   (382 words)

 Chapter IIINVESTIG7%   (Site not responding. Last check: 2007-11-06) Prior to the Forties integral equations with Cauchy principal value type integrals occurred occasionally during investigations of applied problems. Instead one important group of their investigations which made a great impact on the development of the theory of singular integral equations and boundary value problems of the analytic function theory will be presented, namely those linked to ideas outlined by T. Carleman [2]. Using the Sokhotski-Plemelj formulae for the boundary values of the Cauchy type integral (2.4), a one to one correspondence is established between the sets of solutions of.(2.3) and the boundary value problem of analytic function theory kr.cs.ait.ac.th /~radok/mus/mus21.htm   (1201 words)

 Bernard Bialecki - Publications Sinc-Nyström method for numerical solution of one-dimensional Cauchy singular integral equation given on a smooth arc in the complex plane, B. A SINC quadrature subroutine for Cauchy principal value integrals, B. Bialecki and P. Keast, J. Comp. Sinc quadratures for Cauchy principal value integrals, NATO Advanced Research Workshop on Numerical Integration, T.O. Espelid and A. Genz, editors, Kluwer Academic Publishers, 1992, 81-92. www.mines.edu /~bbialeck/publications.html   (1688 words)

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