These notes provide a quick and brief introduction to Fourier Series. The emphasis is not only on the mathematics but also on the history of the subject, its importance, its applications and its place in the rest of science.

I first learnt about Fourier Series as a student of physics. Together with several other assorted topics, they formed a ragbag course called Mathematical Physics from which, when the time came, real physics courses would pick what they wanted. A little later, as a student of mathematics I came across Fourier Series in the middle of a course on Mathematical Analysis. On each occasion, my teachers and my books (all good) managed to keep a secret which I learnt later. Fourier Series are not just tools for the physicist and examples for the mathematician. They are directly responsible for the development of nearly one half of mathematical analysis over the last two centuries.

These notes have been consciously designed to reveal this aspect of the subject and something more. The development of Fourier Series is illustrative of a recurrent pattern in modern science. I hope the reader will see this pattern emerge from our discussion.

This book can be used by a variety of students. Mathematics students at the third year undergraduate level should be able to follow most of the discussion. Typically, such students may have had their first course in Analysis (corresponding to Chapters 1–8 of Principles of Mathematical Analysis by W. Rudin) and have a good working knowledge of complex numbers and basic differential equations.