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On a generalization of Chen ’ s iterated integrals
| Content Provider | Semantic Scholar |
|---|---|
| Author | Joyner, S. M. |
| Copyright Year | 2008 |
| Abstract | Chen’s iterated integrals may be generalized by interpolation of functions of the positive integer number of times which particular forms are iterated in integrals along specific paths, to certain complex values. These generalized iterated integrals satisfy both an additive and a (non-classical) multiplicative iterative property, in addition to a comultiplication formula. This theory is developed in the first part of the paper, after which various applications are discussed, including the expression of certain zeta functions as complex iterated integrals (from which an obstruction to the existence of a contour integration proof of the functional equation for the Dedekind zeta function emerges); an elegant reformulation of a result of Gel’fand and Shilov in the theory of distributions which gives a way of thinking about complex iterated derivatives; and a direct topological proof of the monodromy of polylogarithms. 1 0 Introduction The iterated integrals of K.-T. Chen arise in arithmetic situations, a famous example of which is the occurrence of the polyzeta values (also called multiple zeta values) as periods relating two distinct rational structures on the mixed Hodge structure which comprises the Hodge realization of the motivic fundamental group of P\{0, 1,∞} with tangential base-point −→ 01 . In this paper, it is shown that more general objects, including the polyzeta functions themselves, may be viewed as iterated integrals of a sort generalizing the notion introduced by Chen, and the eventual hope is that such objects could thereby also acquire further arithmetic significance. A very general formulation of these iterated integrals is presented in the first section of the paper, in which it is shown that formal generalizations of the antipode and product formulas satisfied by Chen’s integrals may be ideated and then exploited to define complex iterated integrals along paths for which it is possible to prove a certain iterative property. In particular, whenever the relevant integrals converge, and the necessary iterative property may be established, then for differential 1-forms α and β on some differential manifold M on which γ is a piece-wise smooth path, we define |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://arxiv.org/pdf/0801.0023v2.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Converge Distribution (mathematics) Entity–relationship model Generalization (Psychology) Interpolation Iteration Iterative method Obstruction Positive integer Utility functions on indivisible goods manifold |
| Content Type | Text |
| Resource Type | Article |
