Tauberian theory : a century of developments Tauberian theory compares summability methods for series and integrals helps to decide when there is convergence and provides asymptotic and remainder estimates. The author shows the development of the theory from the beginning and his expert commentary evokes the excitement surrounding the early results. He shows the fascination of the difficult Hardy Littlewood theorems and of an unexpected simple proof and extolls Wieners breakthrough based on Fourier theory. There are the spectacular high indices theorems and Karamatas regular variation which permeates probability theory. The author presents Gelfands elegant algebraic treatment of Wiener theory and his own distributional approach. There is also a new unified theory for Borel and circle methods. The text describes many Tauberian ways to the prime number theorem. A large bibliography and a substantial index round out the book. TOC:The Hardy Littlewood Theorems. Wieners Theory. Complex Tauberian Theorems. Karamatas Heritage: Regular Variation. Extensions of theClassical Theory. Borel Summability and General Circle Methods. Tauberian Remainder Theory. References. Index