Category Archives: Asymptotic Analysis

We consider the asymptotic evaluation of the integral transform $\int_0^\infty f(x) \, \sin^n(\lambda x)/x^n \,\text{d} x$ of an exponential type function $f(x)$ of type $\tau>0$, for large values of the parameter $\lambda$, where $n$ is a positive integer. We refer to this integral as the Sinc transform. Under the condition that $f(x)$ is even with… Read More »

We devise a novel resummation prescription based on the method of finite-part integration [Galapon EA. 2017 Proc. R. Soc A 473, 20160567. (doi:10.1098/rspa.2016.0567)] to perform a constrained extrapolation of the divergent weak-field perturbative expansion for the Heisenberg–Euler Lagrangian to the non-perturbative strong magnetic and electric field regimes. In the latter case, the prescription allowed us… Read More »

We perform an asymptotic evaluation of the Hankel transform, $\int_0^{\infty}J_{\nu}(\lambda x) f(x)\mathrm{d}x$, for arbitrarily large $\lambda$ of an entire exponential type function, $f(x)$, of type $\tau$ by shifting the contour of integration in the complex plane. Under the situation that $J_{\nu}(\lambda x)f(x)$ has an odd parity with respect to $x$ and the condition that the… Read More »

The one-sided and full Hilbert transforms are evaluated exactly by means of the method of finite-part integration [E.A. Galapon, Proc. Roy. Soc. A \textbf{473}, 20160567 (2017)]. In general, the result consists of two terms—the first is an infinite series of finite part of divergent integrals, and the second is a contribution arising from the singularity… Read More »