This book is meant neither as a drill book for the successful nor as a lifebelt for the unsuccessful student. Rather, it is intended as a shop window for some of the ideas, techniques and elegant results of Fourier analysis.

I have tried to write a series of interlinked essays accessible to a student with a good general background in mathematics such as an undergraduate at a British university is supposed to have after two years of study. If the reader has not covered the relevant topic, say contour integration or probability, then she can usually omit, or better, skim through any chapters which involve this topic without impairing her ability to cope with subsequent chapters.

It is a consequence of the plan of this book that nothing is done in great depth or generality. If the reader wants to acquire facility with the Laplace transform or to study the L2 convergence of the Fourier series of an L2 function she must look elsewhere. It is very much easier to acquire a skill or to generalise a theorem when one is under the pressure of immediate necessity than when one is told that such a skill or generalisation might just possibly come in useful some day.

Another consequence is that, although anything specifically presented as a proof or statement of a result is intended to meet the pure mathematician's criteria for accuracy, the rigour of the accompanying discussion will vary according to the subject discussed.