On analytic continuation of various multiple zetafunctions kohji matsumoto abstract in this article we describe the development of the problem of analytic continuation of multiple zetafunctions. A famous unique continuation result from complex analysis says that the analytic. The reason it makes sense to speak of the analytic continuation is the following uniqueness property, which is an immediate consequence of the identity theorem. The power series expansion of the inverse function of an analytic function can be determined using the lagrange inversion theorem. Does this infinite sum provide a new analytic continuation. From the point of view of analytic continuation, the most natural question.
U c are analytic and they agree on any closed disc or. Chapter 9 analytic continuation for every complex problem, there is a solution that is simple, neat, and wrong. Analytic continuation is a technique to extend the domain of a given analytic function. The craziest things what i can construct is a finite number of essential singularities. On analytic continuation of various multiple zetafunctions. A cannonball drag model is utilized and the problem is solved with the. Analytic continuation by means of the lr,tsummability. On the numerical analytic continuation of power series with. The basic idea is to approximate the solution with a power series. Once the power series at s0 are calculated to the desired order, the problem of calculating them at s1 becomes one of analytic continuation. An imprecise version of riemannsexistencetheorem is that these describe the.
First, we present a new class of sets differing from domains, allowing efficient summation by those methods. The process of extending a power series representation of an analytic function is called analytic continuation. Taylor and laurent expansions analytic continuation. In this paper necessary and sufficient conditions for unitarizability are given in the. The approach to analytic functions via power series is the socalled weierstrass approach. Mod06 lec21a analytic continuation along paths via power series part a. The stepwise continuation technique may, however, come up against difficulties. Suppose further that a 1 a 2 and a 0 a 2 are analytic functions. Firstly, an intermediate step involved the identification of a non convergent series with a finite result, which cannot work for all harmonic sums. Moving from this basis back onto the real axis, known as analytic continuation, is an illposed. If is the complex plane or, more generally, the complex space, this process of analytic continuation can be described more simply. It is perhaps more instructive to take a step back to real analytic functions and taylor series, and to see why complex numbers is the natural setting.
Mod06 lec19 the idea of a direct analytic continuation or an analytic extension. Lagrange inversion formula and lagrangeburmann inversion formula. On analytic continuation of the power series outside of the. The analytic function t exists for all complex numbers except z 1 and, inside the unit disk fz2c jjzj power series in 1.
Which power series are proven to not possess an analytic continuation beyond the. These may have an essentially topological nature, leading to inconsistencies. It should be strongly remarked that this does not have anything in common with the techniques of homotopic continuation. How does one motivate the analytic continuation of the. This paper explores several methods of accelerating the convergence of these power series andor providing analytic continuation and distinguishes between those that are backed by the theoretical convergence guarantee of stahls theorem i. The reason it makes sense to speak of the analytic continuation. We begin with the general power series solution method.
Insights the analytic continuation of the lerch and the zeta functions comments. A canonical element is a pair where and is a power series centred at the. Mod06 lec21a analytic continuation along paths via power. Lacunary summability and analytic continuation of power series.
Approximation by entire functions and analytic continuation. The sum of a power series with a positive radius of convergence is an analytic function at every point in the interior of the disc of convergence. How to calculate a cube root using the division method. Power series and analytic continuations numericana. The analytic continuation from the imaginary axis to the complex plane is fz z sinh z cos z sin z. Even though it could be stated simply and precisely as a theorem, doing so would obscure many of the subtleties and how remarkable it is. Power series are useful in analysis since they arise as taylor series. Secondly, the analytic continuation only works when k 1. A novel analytic continuation power series solution for the. Nov 04, 2008 use the power series method to find the general solution to. Aug 12, 2015 mod06 lec21a analytic continuation along paths via power series part a.
Homotopy is powerful since it only makes use of the concept. Pdf on analytic continuation of the power series outside of the. Pdf in this paper the twobody problem with atmospheric drag is considered. Power series method of analytic continuation and natural boundary. This is the second in a series of papers on the analytic continuation of the holomorphic discrete series. In lay terms, analytic continuation is a method to extend the domain of an analytic function the xvalues for which the function converges and thus remains analytic by a stepwise method and. Analytic continuation and summabilty of power series. Abstract in many condensed matter problems, calculations are done on the imaginary axis using a wick rotation. For example you could ask if it is possible to extend the definition of the sin function from the real line to all complex numbers. Instability, which causes difficulties for finite difference solutions, does not affect the power series solutions. There is a close relation between the growth of this entire function and analytic continuation of the power series. By means of analytic continuation, starting from a representation of a function by any one power series, any number of other power series can be found which. The principle of analytic continuation is one of the most essential properties of holomorphic functions. Chernsimons theory with complex gauge group was studied extensively by gukov in gu.
If i have a function f defined by a power series limited by a pole at a say, and i analytically continue the function with another suitably located series then is it possible for the region of validity of second series. Analytic continuation by means of the lr,tsummability transform stanley d. Efficient analytic continuation of power series by matrix. Citeseerx document details isaac councill, lee giles, pradeep teregowda. In the case of one indeterminate bounded newton interpolating series are studied as a generalization of strictly convergent power series. In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the. Mod06 lec21b analytic continuation along paths via power. For this theorem gives us information about the analytic continuation of az,oanzn provided we can find a way of factoring the coefficients. If we assume that a solution of a di erential equation is written as a power series, then perhaps we can use a method reminiscent of undetermined coe cients. Analytic functions are very rigid in the sense that they are entirely determined by their power series expansion at a point, using the process of analytic continuation. Analytic continuation of harmonic sums sciencedirect. Analytic continuation at firstorder phase transitions. Analytic method analytic proposition closure instruction knowledge base.
The basic idea to finding a series solution to a differential equation is to assume that we can write the solution as a power series in the form, yx. Analytic continuation often succeeds in defining further values of a function, for example, in a new region. Now, since power series are functions of x and we know that not every series will in fact exist, it then makes sense to ask if a power series will exist for all x. Power series solution of a differential equation we conclude this chapter by showing how power series can be used to solve certain types of differential equations. Analytic continuation, applied to these examples, would have produced more accurate. The method utilizes kinematic transformation scalar variables differentiated to arbitrary order to generate the required power series coefficients. Each of the power series obtained in this way is convergent only on the circular region of the complex plane centered on z0 and extending to the nearest pole of the function. Mod06 lec21b analytic continuation along paths via power series part b. The riemann function is something called the analytic continuation of the power series. The current work builds on these earlier results by extending the analytic power series approach to trajectory calculations for twobody and j2j6 gravity perturbation terms. Whenever there exists a direct analytic continuation of f1,d1 into a domain.
On the numerical analytic continuation of power series. Introduction the notion of an analytic method is widely used but rarely characterized. Slice of the disk of convergence with its apex on the bounbary. We say that a power series converges for x c if the series.
Power series solution of differential equations wikipedia. Pdf on analytic continuation of the power series outside of. In this paper we discuss some problems on efficient analytic continuation of power series in the complex plane by means of summability methods, given by universal matrices. Its also worth noting that many power series possess a natural boundary, i. This question is answered by looking at the convergence of the power series. Bieberbach, analytische fortsetzung, springer 1955, or v. Chernsimons theory, analytic continuation and arithmetic 339 remark 1. The power series method calls for the construction of a power series solution.
The resulting series can be used to study the solution to problems for which direct calculation is di cult. Power series holomorphic function entire function analytic continuation imaginary axis these keywords were added by machine and not by the authors. And is there a simple general method to get an analytic continuation of a divergent power series. Citeseerx efficient analytic continuation of power series. A truncated series forms a closed approximation of the solution which can be evaluated at any point in the region where the series converges. Oct 12, 2019 inspired by the original developments of recursive power series by means of lagrange invariants for the classical twobody problem, a new analytic continuation algorithm is presented and studied. If we assume that a solution of a di erential equation is written as a power series. The analytic continuation by duality acd method has found applications in lowenergy qcd phenomenology as well as in technicolor models. 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. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Pdf the analytic continuation of the discrete series.
Power series method the power series method can be used to solve odes with variable coe cients. The series solution, with its great accuracy, permits study of the analytic. Apr 01, 2004 lacunary summability and analytic continuation of power series a result on the summability of the geometric is proved which has farreaching consequences with respect to general analytic continuation of power into their mittaglefflerstar, to overconvergence as well as to the universal behavior of trigonometric in the sense of mensov. Theoretical convergence guarantees versus numerical. Analytic continuation sometimes called simply continuation provides a way of extending the domain over which a complex function is defined. To phrase the principle of analytic continuation differently. What the riemann zeta function at s 1 consequently is can be expressed more concretely by referring to riemanns functional equation. Decimated power series are equal to finite sums involving roots of unity.
In section 4, we consider these conditions in the particular case of layered spheroids, compare our results with available numerical calculations, concern the question on singularities of the analytic continuations of wave fields in presence of a spheroidal particle that is far from being clear and that gave rise to controversial results of the earlier analysis. In mathematics, a power series in one variable is an infinite series of the form. Pdf analytic continuation of appells hypergeometric series. The classical hadamard theoremf furnishes one method of doing this for functions of one variable. This paper aims at providing a basic theoretical framework for computations with computable analytic functions and e. So, here it is to make the question complete before it vanishes into mo. Two, even if the series does converge to an analytic function in some region, that region may have a natural boundary beyond which analytic continuation is impossible. There is a property complexvalued functions can have called being analytic it has to do with having a power series representation, something youll learn about in calc 2, and one of the properties of analytic functions is this. Which power series are proven to not possess an analytic continuation beyond the radius of convergence.
Pdf in current work we discuss the issue of the analytic continuation of a power series along a logarithmic spiral outside of the convergence. It can be done whenever there are only isolated singular points. The riemann zeta function at s1 does not expand to the power series that we used to approximate it. Mod06 lec19 the idea of a direct analytic continuation. Let an zn be a power series with finite radius of no. If a 2 is zero for some z, then the frobenius method, a. Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine e. The basic idea is to approximate the solution with a power series of the form. In both cases one series about one endpoint converged considerably faster than the other. The most common application is to a complex analytic function determined near a point by a power series 1 such a power series expansion is in general valid only within its radius of convergence. In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function.
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