East Asian J. Appl. Math., 9 (2019), pp. 409-423.

Published online: 2019-06

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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$). We prove that the
Fourier series converges everywhere within the largest strip centered on the real axis
which is singularity-free, here |$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 $ℑ$($z$) = ±$πχ$, 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.

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$). We prove that the
Fourier series converges everywhere within the largest strip centered on the real axis
which is singularity-free, here |$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 $ℑ$($z$) = ±$πχ$, 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.

*East Asian Journal on Applied Mathematics*.

*9*(3). 409-423. doi:10.4208/eajam.121218.180419