2014 Reprint of 1962 Edition. Full facsimile of the original edition. Not reproduced with Optical Recognition Software. The present volume is an introduction to Fourier series and their use in solving boundary value problems of mathematical physics. The text treats expansions in Fourier series, general orthogonal expansions, convergence of Fourier series, operations with Fourier series, double Fourier series, Fourier integrals and transforms, Bessel functions and Fourier-Bessel series, the eigenfunction method and its use in solving boundary value problems of mathematical analysis, applications to vibrating systems and heat flow problems. Every chapter moves clearly from topic to topic and theorem to theorem, with many theorem proofs given. A total of 107 problems will be found at the ends of the chapters, including many specially added to this English-language edition, and answers are given at the end of the text. Tolstov was one of the foremost mathematicians of the former Soviet Union.