Category Archives: Mathematical Physics

“The problem of missing terms in term by term integration involving divergent integrals” E. A. Galapon (2016) arXiv:1606.09382

By | July 1, 2016
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Term by term integration is one of the frequently used methods in evaluating integrals and in constructing asymptotic expansions of integrals. It typically involves expanding the integrand or a factor of it and then interchanging the order of summation and integration. Sometimes this formal manipulation leads to an infinite series involving divergent integrals. This calls… Read More »

The analytic principal value by way of the Cauchy-Plemelj-Fox theorem

By | May 5, 2016
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In the publication found here or here, I introduced the concept of analytic principal value (APV) to allow meaningful assignment of values to the class of divergent integrals given by (1)   with . The basic assumption in the definition of the APV is that the function has a complex extension , obtained by replacing… Read More »

Errata: The Cauchy principal value and the Hadamard finite part integral as values of absolutely convergent integrals (E.A. Galapon, J. Math. Phys. 57, 033502 (2016))

By | April 29, 2016
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Equations 30 and 31 should read as follows: (1)   (2)   In the original paper, the correct index in the third terms of the right-hand sides of equations 30 and 31 was erroneously encoded as . The post has been appropriately updated. Thanks to Avram Guiterrez, Kenneth Jhon Remo and Aster Niel Abellano (PUP… Read More »

The Cauchy principal value and the Hadamard finite part integral as values of absolutely convergent integrals (E.A. Galapon, J. Math. Phys. 57, 033502 (2016))

By | March 31, 2016
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Abstract The divergent integral , for and , is assigned, under certain conditions, the value equal to the simple average of the contour integrals , where () is a path that starts from and ends at , and which passes above (below) the pole at . It is shown that this value, which we refer… Read More »

The Cauchy principal value and the finite part integral as values of absolutely convergent integrals (E.A. Galapon , arXiv:1512.01323)

By | December 7, 2015
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Abstract The divergent integral , for and , is assigned, under certain conditions, the value equal to the simple average of the contour integrals , where () is a path that starts from and ends at , and which passes above (below) the pole at . It is shown that this value, which we refer… Read More »

Integration by analytic extension

By | October 4, 2015
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Let us consider two complex valued analytic functions, and , with respective domains, and . That a function is analytic means that it is differentiable in its domain except perhaps at some isolated points, such at poles. Let us suppose that the domains of the functions have non-trivial intersection, , which is open. If it… Read More »

Divergent series just got more convergent (Talk given at the SPP-2015 National Conference)

By | September 24, 2015
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Divergent infinite series arise naturally in many areas of physics. They occur, for example, in perturbative solutions to differential equations of mathematical physics. Only in few special cases that closed-form solutions can be found that, in most cases, one inevitably resorts to perturbative solutions which are more likely to diverge than to converge. Divergence, however,… Read More »

Who is afraid of divergent integrals?

By | July 29, 2015
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We, physicists, are obviously not afraid. Our literature, from our notes to our papers to our books, is replete with them. We are not at all embarrassed by our seemingly wanton disregard to mathematical rigor in our mathematical acrobatics that often lead to our ill-defined or infinite-valued divergent integrals. Our enduring affair with such almost… Read More »