This book is based on a two-semester sequence of courses taught to incoming graduate students at the University of Illinois at Urbana-Champaign, primarily physics students but also some from other branches of the physical sciences. The courses aim to introduce students to some of the mathematical methods and concepts that they will find useful in their research. We have sought to enliven the material by integrating the mathematics with its applications. We therefore provide illustrative examples and problems drawn from physics. Some of these illustrations are classical but many are small parts of contemporary research papers. In the text and at the end of each chapter we provide a collection of exercises and problems suitable for homework assignments. The former are straightforward applications of material presented in the text; the latter are intended to be interesting, and take rather more thought and time.

We devote the first, and longest, part (Chapters 1–9, and the first semester in the classroom) to traditional mathematical methods. We explore the analogy between linear operators acting on function spaces and matrices acting on finite-dimensional spaces, and use the operator language to provide a unified framework for working with ordinary differential equations, partial differential equations and integral equations. The mathematical prerequisites are a sound grasp of undergraduate calculus (including the vector calculus needed for electricity and magnetism courses), elementary linear algebra and competence at complex arithmetic. Fourier sums and integrals, as well as basic ordinary differential equation theory, receive a quick review, but it would help if the reader had some prior experience to build on. Contour integration is not required for this part of the book.