C.D. Tica and E.A. Galapon, “Continuation of the Stieltjes Series to the Large Regime by Finite-part Integration,” https://arxiv.org/abs/2302.03891

By | March 5, 2023
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A prescription is devised to utilize a novel convergent expansion in the strong-asymptotic regime of the Stieltjes integral and its generalizations [Galapon E.A Proc.R.Soc A 473, 20160567(2017)] to sum the associated divergent series of Stieltjes across all asymptotic regimes. The novel expansion makes use of the divergent negative-power moments which were treated as Hadamard’s finite… Read More »

P.J.D. Blancas and E.A. Galapon, “Finite-part integration of the Hilbert transform,” https://arxiv.org/abs/2210.14462

By | October 27, 2022
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Finite-part integration is a recently introduced method of evaluating well-defined integrals using the finite-part of divergent integrals [E.A. Galapon, \textit{Proc. Roy. Soc. A} \textbf{473}, 20160567 (2017)]. Here finite-part integration is applied in the exact evaluation of the one-sided, $\operatorname{PV}\!\int_0^{\infty} h(x) (\omega-x)^{-1}\mathrm{d}x$, and full, $\mathrm{PV}\!\int_{-\infty}^{\infty}h(x) (\omega-x)^{-1}\mathrm{d}x$, Hilbert transforms, together with the reductions of the later transform… Read More »

P.C.M. Flores and E.A. Galapon, “Instantaneous tunneling of relativistic massive spin-0 particles,” https://arxiv.org/abs/2207.09040

By | July 20, 2022
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The tunneling time problem earlier studied in Phys. Rev. Lett. \textbf{108}, 170402 (2012) using a non-relativistic time-of-arrival (TOA) operator predicted that tunneling time is instantaneous implying that the wavepacket becomes superluminal below the barrier. The non-relativistic treatment raises the question whether the superluminal behavior is a mere non-relativistic phenomenon or an an inherent quantum effect… Read More »

D.A.L. Pablico and E.A. Galapon, “Quantum corrections to the Weyl quantization of the classical time of arrival,” https://arxiv.org/abs/2205.08694

By | May 19, 2022
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A time of arrival (TOA) operator that is conjugate with the system Hamiltonian was constructed by Galapon without canonical quantization in [J. Math. Phys. \textbf{45}, 3180 (2004)]. The constructed operator was expressed as an infinite series but only the leading term was investigated which was shown to be equal to the Weyl-quantized TOA-operator for entire… Read More »

A family of integral representations for the reciprocal of Pi

By | May 11, 2022
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Theorem 1. The reciprocal of $\pi$ assumes the following family of integral representations \begin{equation} \frac{1}{\pi}= \frac{\left[\prod_{k=1}^{m+n}(2k-1)\right]^2 \lambda^{2m+1}}{(2m)!\, 2^{4m+2n}} \int_0^{\infty} \frac{x^{2m} J_{2m+1}(\lambda x)}{\Gamma\left(m+ n+\frac{1}{2} + x\right) \Gamma\left(m+n+\frac{1}{2}-x\right)}\,{\rm d}x \end{equation} \begin{equation} \frac{1}{\pi}= \frac{\left[\prod_{k=1}^{m+n}(2k-1)\right]^2 \lambda^{2m+2}}{(2m+1)!\, 2^{4m+2n+1}} \int_0^{\infty} \frac{x^{2m+1} J_{2m+2}(\lambda x)}{\Gamma\left(m+n+ \frac{1}{2} + x\right) \Gamma\left(m+n+\frac{1}{2}-x\right)}\,{\rm d}x \end{equation} for $m=0, 1, 2, \dots$, $n=1, 2, 3,\dots$ and $\lambda\geq\pi$, where $J_n(x)$… Read More »

E.A. Galapon, “Finite-part integral representation of the Riemann zeta function at odd positive integers and consequent representations,” https://arxiv.org/abs/2203.11342

By | March 23, 2022
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The values of the Riemann zeta function at odd positive integers, $\zeta(2n+1)$, are shown to admit a representation proportional to the finite-part of the divergent integral $\int_0^{\infty} t^{-2n-1} \operatorname{csch}t\,\mathrm{d}t$. Integral representations for $\zeta(2n+1)$ are then deduced from the finite-part integral representation. Certain relations between $\zeta(2n+1)$ and $\zeta'(2n+1)$ are likewise deduced, from which integral representations for… Read More »

P.C.M Flores and E.A. Galapon, “Relativistic free motion time of arrival operator for massive spin-0 particles with positive energy,” https://arxiv.org/abs/2203.00898

By | March 4, 2022
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A relativistic version of the Aharonov-Bohm time of arrival operator for spin-0 particles was constructed by Razavi in [Il Nuovo Cimento B \textbf{63}, 271 (1969)]. We study the operator in detail by taking its rigged Hilbert space extension. It is shown that the rigged Hilbert space extension of the operator provides more insights into the… Read More »

R.A.E. Farrales, H.B. Domingo and E.A. Galapon, “Conjugates to One Particle Hamiltonians in 1-Dimension in Differential Form,” https://arxiv.org/abs/2201.05777

By | March 4, 2022
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A time operator is a Hermitian operator that is canonically conjugate to a given Hamiltonian. For a particle in 1-dimension, a Hamiltonian conjugate operator in position representation can be obtained by solving a hyperbolic second-order partial differential equation, known as the time kernel equation, with some boundary conditions. One possible solution is the time of… Read More »

E.A. Galapon, “Regularized limit, analytic continuation and finite-part integration,” https://arxiv.org/abs/2108.02013

By | March 4, 2022
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Finite-part integration is a recent method of evaluating a convergent integral in terms of the finite-parts of divergent integrals deliberately induced from the convergent integral itself [E. A. Galapon, Proc. R. Soc., A 473, 20160567 (2017)]. Within the context of finite-part integration of the Stieltjes transform of functions with logarithmic growths at the origin, the… Read More »

L. Villanueva and E.A. Galapon, “Finite-part integration in the presence of competing singularities: Transformation equations for the hypergeometric functions arising from finite-part integration”

By | December 5, 2020
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Finite-part integration is a recently introduced method of evaluating convergent integrals by means of the finite part of divergent integrals [E.A. Galapon, {\it Proc. R. Soc. A 473, 20160567} (2017)]. Current application of the method involves exact and asymptotic evaluation of the generalized Stieltjes transform $\int_0^a f(x)/(\omega + x)^{\rho} \, \mathrm{d}x$ under the assumption that… Read More »