@Article{EAJAM-9-3,
author = {},
title = {The Breakdown of Darboux's Principle and Natural Boundaries for a Function Periodised from a Ramanujan Fourier Transform Pair},
journal = {East Asian Journal on Applied Mathematics},
year = {2019},
volume = {9},
number = {3},
pages = {409--423},
abstract = {<p style="text-align: justify;">Darboux&#39;s Principle asserts that a power series or Fourier coefficient $a$<sub>$n$</sub> for an
analytic function $f$($z$) is approximated as $n$ → $∞$ by a sum of terms, one for each singularity of <span style="text-align: justify;">$f$($z$)</span> in the complex plane. This is crucial to understanding why Fourier series
converge rapidly or slowly, and thus crucial to Fourier numerical methods. We partially
refute Darboux&#39;s Principle by an explicit counterexample constructed by applying the
Poisson Summation Theorem to a Fourier Transform pair found explicitly by Ramanujan. The Fourier coefficients show a geometric rate of decay proportional to exp(−$πχ$$n$)
multiplied by sin($φ$) where the “phase&quot; is $φ$ = $π$$χ$<sup>2</sup>$n$<sup>2</sup>&nbsp;mod (2$π$). We prove that the
Fourier series converges everywhere within the largest strip centered on the real axis
which is singularity-free, here |$\mathcal{I}$($z$)| &lt; $πχ$. We present strong evidence that the boundaries of the strip of convergence are natural boundaries. Because the function $f$($z$) is
singular everywhere on the lines $\mathcal{I}$($z$) = ±$πχ$, there is no simple way to extrapolate
the asymptotic form of the Fourier coefficients from knowledge of the singularities, as is
possible through Darboux&#39;s Theorem when the singularities are isolated poles or branch
points.</p>},
issn = {2079-7370},
doi = {https://doi.org/10.4208/eajam.121218.180419},
url = {https://global-sci.com/article/82612/the-breakdown-of-darbouxs-principle-and-natural-boundaries-for-a-function-periodised-from-a-ramanujan-fourier-transform-pair}
}