Power series from a computational point of view (Q1097363)

The author's objective is to provide as much theory as students would need in order to calculate values of functions defined by power series (with complex coefficients) to a given precision. To this end, he provides rigorous epsilon-delta proofs (taking \(\epsilon =10^{-6}\) for concreteness); his other objective is to teach students how to construct such proofs. The theory begins with Taylor series and goes as far as analytic continuation and the monodromy theorem. The level is rather uneven: the material on Taylor's theorem, sequences and series could have been copied from a calculus book, whereas the rest of the book is a serious presentation of those (and only those) parts of complex analysis that the author needs. As applications, there are the basic theorems on solvability of linear differential equations, and of polynomial equations, with analytic coefficients. The book is actually a set of lecture notes, reproduced photographically from type-written copy, and exhibits the worst features of this genre: incomplete proofs and lack of motivation. Some of the notation is confusing, particularly the use of f n to mean both an nth derivative and an nth power, sometimes both in the same formula. The intended audience would have a hard time following the exposition, from the beginnging onward, without a mentor to supply the background and rational for the proofs which were presumably given orally in the original lectures. For example, Goursat's proof of Cauchy's theorem is presented with neither explanation of the plan of the proof nor any mention of Goursat. The book is full of distracting small errors; it seems not to have been proof-read by anybody; one wonders whether the editors of the series look at what they accept.