McGraw-Hill, 1980. - 187 Pages.
Over forty years have elapsed since the appearance of Turing's brilliant paper titled "On Computable Numbers, with an Application to the Entscheidungsproblem. " That work contained the first definition of the computable numbers. In the years since, through the efforts of many mathematicians, an appropriate way of dealing with these numbers has arisen that quite properly may be called an analysis, and the designations "recursive analysis, " "constructive analysis, " and "computable analysis" all have been used. In recent years the development of this analysis has been carried out mainly in Easte European countries, with the Russian mathematicians Markov, Ceitin, and Zaslavskii making many of the early fundamental contributions.
As explained in Chapter 1, it is entirely possible to view the analysis as a subanalysis of ordinary real analysis. This subanalysis provides insight into the many difficulties encountered when numerical computations are attempted, difficulties that often go unrecognized in real analysis.
In this text I attempt to give an elementary, self-contained presentation of computable analysis. Familiarity of the reader with the material normally covered in a university course on advanced calculus is sufficient for understanding all chapters except the last, where some acquaintance with complex analysis is presupposed. In the back of the book the section "Notes on References" gives sources for the more important theorems resented.
Over forty years have elapsed since the appearance of Turing's brilliant paper titled "On Computable Numbers, with an Application to the Entscheidungsproblem. " That work contained the first definition of the computable numbers. In the years since, through the efforts of many mathematicians, an appropriate way of dealing with these numbers has arisen that quite properly may be called an analysis, and the designations "recursive analysis, " "constructive analysis, " and "computable analysis" all have been used. In recent years the development of this analysis has been carried out mainly in Easte European countries, with the Russian mathematicians Markov, Ceitin, and Zaslavskii making many of the early fundamental contributions.
As explained in Chapter 1, it is entirely possible to view the analysis as a subanalysis of ordinary real analysis. This subanalysis provides insight into the many difficulties encountered when numerical computations are attempted, difficulties that often go unrecognized in real analysis.
In this text I attempt to give an elementary, self-contained presentation of computable analysis. Familiarity of the reader with the material normally covered in a university course on advanced calculus is sufficient for understanding all chapters except the last, where some acquaintance with complex analysis is presupposed. In the back of the book the section "Notes on References" gives sources for the more important theorems resented.