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Synopsis This book provides a quick introduction to Fourier Series. The author begins by describing one of the problems actually studied by Fourier and how this leads to a differential equation to solve which Fourier introduced his famous series. The mathematical problems this leads to are then discussed rigorously. After that, applications of the theory to four different problems are given. A novel feature of the book is the inclusion of a detailed history of the subject. This would help the student understand why this topic is important. In the revised edition, Chapter 3 has been considerably expanded to include more examples and connections with related topics. A large part of the book can be read by third year B.Sc. Hons students in India and can be used by them to supplement their analysis courses and to sharpen their understanding of concepts like convergence, uniform convergence etc. For some later portions a knowledge of Functional Analysis is required. These parts can be read by M.Sc. and M. Phil students. The book would also be useful for summer schools and refresher courses for students and college teachers. To such an audience all the material can be taught in about one month.