Author Archives: Eric A. Galapon

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 \begin{equation}\label{divergent} \int_a^b\frac{f(x)}{(x-x_0)^{n+1}}\mathrm{d}x, \; \; n=0, 1, 2, \dots . \end{equation}with $f(x_0)\neq 0$. The basic assumption in the definition of the APV is that the function… Read More: The analytic principal value by way of the Cauchy-Plemelj-Fox theorem »

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: $\setCounter{29}$ \begin{eqnarray} \int_a^{x_0-\epsilon} \frac{f(x)}{(x-x_0)^{n+1}}\mbox{d}x &=& -\sum_{k=0}^{n-1} \frac{f^{(k)}(x_0)}{k!(n-k)} \left(\frac{1}{(-\epsilon)^{n-k}} – \frac{1}{(a-x_0)^{n-k}}\right) \nonumber \\ && + \frac{f^{(n)}(x_0)}{n!} \left(\ln \epsilon – \ln(x_0-a)\right)\nonumber \\ && + \sum_{k=n+1}^{\infty} \frac{f^{(k)}(x_0)}{k!(k-n)} \left((-\epsilon)^{k-n}-(a-x_0)^{k-n}\right) \end{eqnarray}\begin{eqnarray} \int_{x_0+\epsilon}^b \frac{f(x)}{(x-x_0)^{n+1}}\mbox{d}x &=& -\sum_{k=0}^{n-1} \frac{f^{(k)}(x_0)}{k!(n-k)} \left(\frac{1}{(b-x_0)^{n-k}} – \frac{1}{\epsilon^{n-k}}\right) \nonumber \\ && + \frac{f^{(n)}(x_0)}{n!} \left(\ln (b-x_0) – \ln\epsilon\right)\nonumber \\ && + \sum_{k=n+1}^{\infty}… Read More: Errata: The Cauchy principal value and the Hadamard finite part… »

Internal one degree of freedom is sufficient to induce decoherence (E. A. Galapon 2016 EPL 113 60007)

By | April 15, 2016
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Abstract Current quantum orthodoxy claims that the statistical collapse of the wave-function arises from the interaction of the measuring instrument with its environment through the phenomenon known as environment-induced decoherence. Here it is shown that there exists a measurement scheme that is exactly decohering without the aid of an environment. The scheme relies on the… Read More: Internal one degree of freedom is sufficient to induce decoherence… »

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 $\int_a^b f(x)(x-x_0)^{-n-1}\mathrm{d}x$, for $-\infty<a<x_0<b<\infty$ and $n=0,1,2,\dots$, is assigned, under certain conditions, the value equal to the simple average of the contour integrals $\int_{C^{\pm}} f(z)(z-x_0)^{-n-1}\mathrm{d}z$, where $C^+$ ($C^-$) is a path that starts from $a$ and ends at $b$, and which passes above (below) the pole at $x_0$. It is shown that… Read More: The Cauchy principal value and the Hadamard finite part integral… »

If you’re losing your grip on quantum mechanics, you may want to read up on this

By | January 8, 2016
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To many students, quantum mechanics is an academically imposed phantom that comes and goes as quickly as the semesters start and end. A primary reason, I believe, is that they find it so far removed from their familiarities. They approach quantum mechanics with the Newtonian mindset which is in accord with their everyday experience. When… Read More: If you’re losing your grip on quantum mechanics, you may… »

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 $\int_a^b f(x)(x-x_0)^{-n-1}\mathrm{d}x$, for $-\infty<a<x_0<b<\infty$ and $n=0,1,2,\dots$, is assigned, under certain conditions, the value equal to the simple average of the contour integrals $\int_{C^{\pm}} f(z)(z-x_0)^{-n-1}\mathrm{d}z$, where $C^+$ ($C^-$) is a path that starts from $a$ and ends at $b$, and which passes above (below) the pole at $x_0$. It is shown that… Read More: The Cauchy principal value and the finite part integral as… »

Integration by analytic extension

By | October 4, 2015
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Let us consider two complex valued analytic functions, $g_1(z)$ and $g_2(z)$, with respective domains, $D_1$ and $D_2$. 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, $D_{12}=D_1\cap D_2\neq \emptyset$, which… Read More: Integration by analytic extension »

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: Divergent series just got more convergent (Talk given at the… »