Activities
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Seminars
| Seminar on Theoretical and Computational Physics:
| | Title | Duality of divergent integrals: theory and applications
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| Speaker | Prof. Eric Galapon
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| Affiliation | National Institute of Physics University of the Philippines, Diliman, Quezon City
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| Date | Tuesday, 24-07-2018
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| Time | 10:00 AM
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| Location | Meeting room, 6th floor, 2H Building, 18 Hoang Quoc Viet, Hanoi
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| Abstract | Divergent integrals arise naturally in many areas of physics. They typically appear from well defined physical problems when the order of two operations involving integration are interchanged, such as integration and summation, limit, and other integration. The divergence indicates that necessary uniformity conditions are not satisfied. Under such circumstance, the mathematician would abandon the interchange but the physicist would proceed because, in many situtions, the interchange is the only option forward. The emerging divergent integrals are then made meaningful by the physicist by regularization, which is a mathematical trick that assigns a definite value to a divergent integral. But does the assigned value lead to the correct result? In this talk we will discuss recent progress in the theory of divergent integrals that directly confronts the question of the correctness of the assigned value to a divergent integral. We will examine examples of the use of regularization by analytic continuation that lead to correct and incorrect results; in the later case, analytic continuation correctly reproduces some terms of the correct result but completely misses out a group of terms. Until recently it is not known why analytic continuation would work in certain problems but not in other problems. The recent discovery that specific regularization of divergent integrals assume a dual property clarifies under certain what conditions that a regularization works or not. The duality does not only identify the conditions but also provides the missing terms when the regularization fails. The talk aims at elucidating this duality of divergent integrals.
References:
E.A. Galapon, “The problem of missing terms in term by term integration involving divergent integrals,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, 20160567 (2017).
E.A. Galapon, “The Cauchy principal value and the Hadamard finite part integral as values of absolutely convergent integrals,” J. Math. Phys. 57, 033502 (2016).
E.A. Galapon, "Integration by divergent integrals: Calculus of divergent integrals in term by term integration,"arxiv: 1709.08173 (under review).
C.D. Tica and E.A. Galapon, “Finite-part integration of the generalized Stieltjes transform and its dominant asymptotic behavior for small values of the parameter. I. Integer orders,” J. Math. Phys. 59, 023509 (2018).
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| Host person | Trịnh Xuân Hoàng
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