Extract from the Preface to the First Edition

THIS book has been designed primarily for the use of first year students at the Universities whose abilities reach or approach something like what is usually described as ‘scholarship standard’. I hope that it may be useful to other classes of readers, but it is this class whose wants I have considered first. It is in any case a book for mathematicians: I have nowhere made any attempt to meet the needs of students of engineering or indeed any class of students whose interests are not primarily mathematical.

I regard the book as being really elementary. There are plenty of hard examples (mainly at the ends of the chapters): to these I have added, wherever space permitted, an outline of the solution. But I have done my best to avoid the inclusion of anything that involves really difficult ideas. For instance, I make no use of the ‘principle of convergence’: uniform convergence, double series, infinite products, are never alluded to: and I prove no general theorems whatever concerning the inversion of limit-operations—I never even define $\frac{{\partial }^{2}f}{\partial x\partial y}$ and $\frac{{\partial }^{2}f}{\partial y\partial x}$. In the last two chapters I have occasion once or twice to integrate a power-series, but I have confined myself to the very simplest cases and given a special discussion in each instance. Anyone who has read this book will be in a position to read with profit Dr Bromwich’s Infinite Series, where a full and adequate discussion of all these points will be found.