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Mittag-Leffler Summation
In mathematics, Mittag-Leffler summation is any of several variations of the Borel summation method for summing possibly divergent formal power series, introduced by Definition Let :y(z) = \sum_^\infty y_kz^k be a formal power series in ''z''. Define the transform \scriptstyle \mathcal_\alpha y of \scriptstyle y by :\mathcal_\alpha y(t) \equiv \sum_^\infty \fract^k Then the Mittag-Leffler sum of ''y'' is given by :\lim_\mathcal_\alpha y( z) if each sum converges and the limit exists. A closely related summation method, also called Mittag-Leffler summation, is given as follows . Suppose that the Borel transform \mathcal_1 y(z) converges to an analytic function near 0 that can be analytically continued along the positive real axis to a function growing sufficiently slowly that the following integral is well defined (as an improper integral). Then the Mittag-Leffler sum of ''y'' is given by :\int_0^\infty e^ \mathcal_\alpha y(t^\alpha z) \, dt When ''α'' = 1 this i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Summation
In mathematics, Borel summation is a summation method for divergent series, introduced by . It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum for such series. There are several variations of this method that are also called Borel summation, and a generalization of it called Mittag-Leffler summation. Definition There are (at least) three slightly different methods called Borel summation. They differ in which series they can sum, but are consistent, meaning that if two of the methods sum the same series they give the same answer. Throughout let denote a formal power series :A(z) = \sum_^\infty a_kz^k, and define the Borel transform of to be its equivalent exponential series :\mathcalA(t) \equiv \sum_^\infty \fract^k. Borel's exponential summation method Let denote the partial sum :A_n(z) = \sum_^n a_k z^k. A weak form of Borel's summation method defines the Borel sum of to be : \lim_ e^\sum_^\infty \frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divergent Series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series :1 + \frac + \frac + \frac + \frac + \cdots =\sum_^\infty\frac. The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme. In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series. A ''summability method'' or ''summation method'' is a partial function from the set of series to values. For example, Cesàro summation assigns Grandi's divergent ser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Power Series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, whose terms are of the form a x^n where x^n is the nth power of a variable x (n is a non-negative integer), and a is called the coefficient. Hence, power series can be viewed as a generalization of polynomials, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the x^n are used only as position-holders for the coefficients, so that the coefficient of x^5 is the fifth ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about ''x''0 converges to the function in some neighborhood for every ''x''0 in its domain. Definitions Formally, a function f is ''real analytic'' on an open set D in the real line if for any x_0\in D one can write : f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + a_3 (x-x_0)^3 + \cdots in which the coefficients a_0, a_1, \dots are real numbers and the series is convergent to f(x) for x in a neighborhood of x_0. Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at any point x_0 in its domain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent. The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of singularities. The case of several complex variables is rather different, since singularities then need not be isolated points, and its investigation was a major reason for the development of sheaf cohomology. Initial discussion Suppose ''f'' is an analytic function defined on a non-empty open subset ''U'' of the complex plane If ''V'' is a larger open subset of containing ''U'', and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Real Axis
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used for either of these, the notation \R_ or \R^ for \left\ and \R_^ or \R^_ for \left\ has also been widely employed, is aligned with the practice in algebra of denoting the exclusion of the zero element with a star, and should be understandable to most practicing mathematicians. In a complex plane, \R_ is identified with the positive real axis, and is usually drawn as a horizontal ray. This ray is used as reference in the polar form of a complex number. The real positive axis corresponds to complex numbers z = , z, \mathrm^, with argument \varphi = 0. Properties The set \R_ is closed under addition, multiplication, and division. It inherits a topology from the real line and, thus, has the structure of a multiplicative topological grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mittag-Leffler Function
In mathematics, the Mittag-Leffler function E_ is a special function, a complex function which depends on two complex parameters \alpha and \beta. It may be defined by the following series when the real part of \alpha is strictly positive: :E_ (z) = \sum_^\infty \frac, where \Gamma(x) is the gamma function. When \beta=1, it is abbreviated as E_\alpha(z) = E_(z). For \alpha=0, the series above equals the Taylor expansion of the geometric series and consequently E_(z)=\frac\frac. In the case \alpha and \beta are real and positive, the series converges for all values of the argument z, so the Mittag-Leffler function is an entire function. This function is named after Gösta Mittag-Leffler. This class of functions are important in the theory of the fractional calculus. For \alpha >0 , the Mittag-Leffler function E_(z) is an entire function of order 1/\alpha, and is in some sense the simplest entire function of its order. The Mittag-Leffler function satisfies the recurrence prope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nachbin's Theorem
In mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is commonly used to establish a bound on the growth rates for an analytic function. This article provides a brief review of growth rates, including the idea of a function of exponential type. Classification of growth rates based on type help provide a finer tool than big O or Landau notation, since a number of theorems about the analytic structure of the bounded function and its integral transforms can be stated. In particular, Nachbin's theorem may be used to give the domain of convergence of the generalized Borel transform, given below. Exponential type A function ''f''(''z'') defined on the complex plane is said to be of exponential type if there exist constants ''M'' and α such that :, f(re^), \le Me^ in the limit of r\to\infty. Here, the complex variable ''z'' was written as z=re^ to emphasize that the limit must hold in all directions θ. Letting α stand for the infimum o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
