THE INTEGRATION OF FUNCTIONS OF A SINGLE VARIABLEby
G H HARDY, M.A.
(Fellow of Trinity College)
We give below the Preface to Hardy's pamphlet:
This pamphlet is intended to be read as a supplement to the accounts of 'Indefinite Integration' given in text-books on the Integral Calculus. The student who is only familiar with the latter is apt to be under the impression that the process of integration is essentially 'tentative' in character, and that its performance depends on a large number of disconnected though ingenious devices. My object has been to do what I can to show that this impression is mistaken, by showing that the solution of any elementary problem of integration may be sought in a perfectly definite and systematic way.
The reader who is familiar with the theory of algebraical functions and algebraical plane curves will no doubt find the treatment in Section V. of the integrals of algebraical functions sketchy and inadequate. I hope, however, that he will bear in mind the great difficulty of presenting even an outline of the elements of so vast a subject in a short space and without presupposing a wider range of mathematical knowledge than I am at liberty to assume.
I have naturally not said much about particular devices which are only useful in special cases, but I have tried to show, where it is possible, how such devices find their place in the general theory. And I would strongly recommend any reader who is not already familiar with the general processes here explained to work through a number of examples (those for instance which have been set in the Mathematical Tripos in recent years) using in each case both the general method and any special method which he may find better adapted to the particular case.
I have borrowed largely from the Cours d'Analyse of Hermite and Goursat, but my greatest debt is to Liouville, who published in the years 1830-40 a series of remarkable memoirs on the general problem of integration which appear to have fallen into an oblivion which they certainly do not deserve. It was Liouville who first gave rigid proofs of whole series of theorems of the most fundamental importance in analysis - that the exponential function is not algebraical, that the logarithmic function cannot be expressed by means of algebraical and exponential functions, and that the standard elliptic, integrals cannot be expressed by algebraical, exponential and logarithmic functions. That such theorems require proof is too often altogether forgotten.
I have added a list of references for the benefit of more advanced readers.
G H H