4 Remains of Mr. John Oldham

A complex analysis approach to Atangana–Baleanu fractional calculus

“…Proof. We showed in Lemma 3.1 that the formulae (34) and (36) are valid extensions of the original ABR and ABC derivatives from 0 < ν < 1 to Re(ν) > 0.…”

“…[8,25,26,27,30,29,38,40] In many applications, it is enough to consider the order of differintegration ν as a real number, or even a number in some finite interval such as [0, 1]. These conventions are used for the most part in various sources including the standard textbooks [32,34]. However, it is also interesting and important to consider fractional extensions of complex analysis.…”

“…A good way of resolving the branch cut issue is to redefine the Riemann-Liouville fractional differintegral in a way more suited to complex analysis. Namely, the following formula, which was first proposed in [33] and further examined in [35,34,36]:…”

“…for f an analytic function [33,34]. The proof for ℜ(ν) < 0 involves collapsing the Hankel contour of integration in (3) onto the branch cut, dividing the integral into three parts, and using the reflection formula for gamma functions to recover (1).…”

An inflammatory match? Public anxiety and political assurance at the wedding of William III and Mary II

“…65 Accordingly, eulogists prayed that she might prove as fertile as her grandmother, Henrietta Maria. 66 Comparing the events of the wedding day to the deliverance of Protestantism from the gunpowder plot, one writer predicted that the couple's offspring would guarantee the preservation of the reformed faith:…”