No abstract

We obtain an exactification of the Poincaré asymptotic expansion (PAE) of the Hankel integral,We find that, for halfinteger orders of the Bessel function, the exactified asymptotic series terminates, so that it gives an exact finite sum representation of the Hankel integral. For other orders, the asymptotic series does not terminate and is generally divergent, but is amenable to superasymptotic summation, i.e. by optimal truncation. For specific examples, we compare the accuracy of the optimally truncated asymptotic series owing to the McClure-Wong distributional method with owing to the Mellin-Barnes integral method. We find that the former is spectacularly more accurate than the latter, by, in some cases, more than 70 orders of magnitude for the same moderate value of b. Moreover, the exactification can lead to a resummation of the PAE when it is exact, with the resummed Poincaré series exhibiting again the same spectacular accuracy. More importantly, the distributional method may yield meaningful resummations that involve scales that are not asymptotic sequences.

One of the best sources of information about our cultural origin are written texts. Often, texts are hidden, sometimes erased, faded away, or written over, sometimes not easily accessible in rolled or folded documents. Due to recent improvements in sensitivity and resolution, spectacular disclosures of rolled hidden texts were possible by X-ray tomography, most of them made out of parchment. However, revealing text on folded manuscripts is even more challenging. Due to the fragile condition of fragments, manual unfolding is often too risky, as it can lead to the total loss of the document. X-ray tomography allows for virtual unfolding and enables non-destructive access to hidden texts. Here, the progress in virtual unfolding is reviewed, focusing on papyri from Elephantine Island near Aswan. The project is a part of the European Research Council's starting grant ELEPHANTINE. Results on unfolding ancient papyrus packages from the papyrus collection of the Musée du Louvre and of the Ägyptisches Museum und Papyrussammlung, Berlin are discussed.