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Perturbation Problem Beyond All Orders
In mathematics, perturbation theory works typically by expanding unknown quantity in a power series in a small parameter. However, in a perturbation problem beyond all orders, all coefficients of the perturbation expansion vanish and the difference between the function and the constant function 0 cannot be detected by a power series. A simple example is understood by an attempt at trying to expand e^ in a Taylor series in \epsilon > 0 about 0. All terms in a naïve Taylor expansion are identically zero. This is because the function e^ possesses an essential singularity at z = 0 in the complex z-plane, and therefore the function is most appropriately modeled by a Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ... -- a Taylor series has a zero radius of convergence. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perturbation Theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In perturbation theory, the solution is expressed as a power series in a small parameter The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of \varepsilon usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, usually by keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction. Perturbation theory is used in a wide range of fields, and reaches its most sophisticated and advanced forms in quantum field theory. Perturbation theory (quantum mechanics) describes the use of this method in quantum mechanics. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, ''c'' (the ''center'' of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewed as an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Function
In mathematics, especially real analysis, a flat function is a smooth function f : \mathbb \to \mathbb all of whose derivatives vanish (mathematics), vanish at a given point x_0 \in \mathbb. The flat functions are, in some sense, the antitheses of the analytic functions. An analytic function f : \mathbb \to \mathbb is given by a convergent series, convergent power series close to some point x_0 \in \mathbb: :f(x) \sim \lim_ \sum_^n \frac (x-x_0)^k . In the case of a flat function, all derivatives vanish at x_0 \in \mathbb, i.e. f^(x_0) = 0 for all k \in \mathbb. This means that a meaningful Taylor series expansion in a neighbourhood (mathematics), neighbourhood of x_0 is impossible. In the language of Taylor's_theorem#Statement, Taylor's theorem, the non-constant part of the function always lies in the remainder R_n(x) for all n \in \mathbb. The function need not be flat at just one point. Trivially, constant functions on \mathbb are flat everywhere. But there are also other, less ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Essential Singularity
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior. The category ''essential singularity'' is a "left-over" or default group of isolated singularities that are especially unmanageable: by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner – removable singularities and poles. In practice some include non-isolated singularities too; those do not have a residue. Formal description Consider an open subset U of the complex plane \mathbb. Let a be an element of U, and f\colon U\setminus\\to \mathbb a holomorphic function. The point a is called an ''essential singularity'' of the function f if the singularity is neither a pole nor a removable singularity. For example, the function f(z)=e^ has an essential singularity at z=0. Alternative descriptions Let \;a\; be a complex number, assume that f(z) is not defined at \;a\; but is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laurent Series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass may have discovered it first in a paper written in 1841, but it was not published until after his death.. Definition The Laurent series for a complex function f(z) about a point c is given by f(z) = \sum_^\infty a_n(z-c)^n, where a_n and c are constants, with a_n defined by a line integral that generalizes Cauchy's integral formula: a_n =\frac\oint_\gamma \frac \, dz. The path of integration \gamma is counterclockwise around a Jordan curve enclosing c and lying in an annulus A in which f(z) is holomorphic (analytic). The expansion for f(z) will then be valid anywhere inside the annulus. The annulus is shown in red ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radius Of Convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal to the radius of convergence, and it is the Taylor series of the analytic function to which it converges. In case of multiple singularities of a function (singularities are those values of the argument for which the function is not defined), the radius of convergence is the shortest or minimum of all the respective distances (which are all non-negative numbers) calculated from the center of the disk of convergence to the respective singularities of the function. Definition For a power series ''f'' defined as: :f(z) = \sum_^\infty c_n (z-a)^n, where *''a'' is a complex constant, the center of the disk of convergence, *''c''''n'' is the ''n''-th comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Expansion
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations by revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function. The most common type of asymptotic expansion is a power series in either positive or negative powers. Methods of generating such expansions include the Euler–Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion. Since a '' convergent'' Taylor series fits the definition of asymptotic expansion as well, the phrase "asymptotic series" usually implies a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perturbation Theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In perturbation theory, the solution is expressed as a power series in a small parameter The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of \varepsilon usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, usually by keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction. Perturbation theory is used in a wide range of fields, and reaches its most sophisticated and advanced forms in quantum field theory. Perturbation theory (quantum mechanics) describes the use of this method in quantum mechanics. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
