TY - JOUR T1 - The Breakdown of Darboux's Principle and Natural Boundaries for a Function Periodised from a Ramanujan Fourier Transform Pair AU - John P. Boyd JO - East Asian Journal on Applied Mathematics VL - 3 SP - 409 EP - 423 PY - 2019 DA - 2019/06 SN - 9 DO - http://doi.org/10.4208/eajam.121218.180419 UR - https://global-sci.org/intro/article_detail/eajam/13159.html KW - Darboux’ Theorem, Fourier Transform, imbricate, Ramanujan, natural boundary AB -

Darboux's Principle asserts that a power series or Fourier coefficient $a$$n$ for an analytic function $f$($z$) is approximated as $n$ → $∞$ by a sum of terms, one for each singularity of $f$($z$) in the complex plane. This is crucial to understanding why Fourier series converge rapidly or slowly, and thus crucial to Fourier numerical methods. We partially refute Darboux's Principle by an explicit counterexample constructed by applying the Poisson Summation Theorem to a Fourier Transform pair found explicitly by Ramanujan. The Fourier coefficients show a geometric rate of decay proportional to exp(−$πχ$$n$) multiplied by sin($φ$) where the “phase" is $φ$ = $π$$χ$2$n$2 mod (2$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$)| < $πχ$. 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's Theorem when the singularities are isolated poles or branch points.