Formal Power Series of Logarithms
Since its beginning with Barrow and Newton, calculus of finite differences has been viewed-whether admittedly or not-as a proper method of computation with special functions. Until the nineteenth century, when the trials of convergence were to be gradually imposed, mathematicians handling difference equations and series involving polynomials and logarithms weren't beset by doubts on the correctness of their manipulations, and actually, their results have seldom turned out to be incorrect, even by the standards of our day. There were, however, some embarrassing exceptions-which mathematicians during this century have chosen largely to ignore. Perhaps the simplest known of those is that the Euler-MacLaurin summation formula. When applied to any “function” aside from a polynomial, this formula gives a divergent series, which nonetheless are often wont to obtain astonishingly good numerical approximations, and which may be used without worrying in formal manipulations. It’s of little help to justify such manipulations by appealing to Poincarit’s definition of an asymptotic expansion. Most Euler-MacLaurin series contain logarithmic terms and other functions growing slower than any polynomial, whose asymptotic expansion consistent with Poincart would equal zero. The suggestion first made by Dubois-Reymond, and later haunted by G. H. Hardy-that the notion of an asymptotic expansion be reinforced by logarithmic scales-has not been developed, neither is it clear how the difficulties of its implementation are to be surmounted, or maybe whether such difficulties are worth surmounting.