This book is an introductory presentation to the theory of local zeta functions. Viewed as distributions, and mostly in the archimedean case, local zeta functions are also called complex powers. The volume contains major results on analytic and algebraic properties of complex powers by Atiyah, Bernstein, I. M. Gelfand, S. I. Gelfand, and Sato. Chapters devoted to $p$-adic local zeta functions present Serre's structure theorem, a rationality theorem, and many examples found by the author. The presentation concludes with theorems by Denef and Meuser. Information for our distributors: Titles in this series are co-published with International Press, Cambridge, MA.
Preliminaries Implicit function theorems and $K$-analytic manifolds Hironaka's desingularization theorem Bernstein's theory Archimedean local zeta functions Prehomogeneous vector spaces Totally disconnected spaces and $p$-adic manifolds Local zeta functions ($p$-adic case) Some homogeneous polynomials Computation of $Z(s)$ Theorems of Denef and Meuser Bibliography Index.