Carslaw H.S. Introduction to the theory of Fourier's series and integrals

London: Macmillan and Co., limited, 1921. — 319 p.This book forms the first volume of the new edition of my book on Fourier's Series and Integrals and the Mathematical Theory of the Conduction of Heat, published in 1906. and now for some time out of print. Since 1906 so much advance has been made in the Theory of Fourier's Series and Integrals, as well as in the mathematical discussion of Heat Conduction, that it has seemed advisable to write a completely new work, and to issue the same in two volumes. The first volume, which now appears, is concerned with the Theory of Infinite Series and Integrals, with special reference to Fourier's Series and Integrals. The second volume will be devoted to the Mathematical Theory of the Conduction of Heat.
No one can properly understand Fourier's Series and Integrals without a knowledge of what is involved in the convergence of infinite series and integrals. With these questions is bound up the development of the idea of a limit and a function, and both are founded upon the modern theory of real numbers. The first three chapters deal with these matters. In Chapter IV. the Definite Integral is treated from Riemann's point of view, and special attention is given to the question of the convergence of infinite integrals. The theory of series whose terms are functions of a single variable, and the theory of integrals which contain an arbitrary parameter are discussed in Chapters V. and VI. It will be seen that the two theories are closely related, and can be developed on similar lines.
The treatment of Fourier's Series in Chapter VII. depends on Dirichlet's Integrals. There, and elsewhere throughout the book, the Second Theorem of Mean Value will be found an essential part of the argument. In the same chapter the work of Poisson is adapted to modern standards, and a prominent place is given to Fejer's work, both in the proof of the fundamental theorem and in the discussion of the nature of the convergence of Fourier5s Series. Chapter IX. is devoted to Gibbs's Phenomenon, and the l&st chapter to Fourier's Integrals. In this chapter the work of Pringsheim, who has greatly extended the class of functions to which Fourier's Integral Theorem applies, has been used.