THE INTEGRATION OF FUNCTIONS OF A SINGLE VARIABLE I. Introduction
Reading edition
The problem considered in the following pages is what is sometimes called the problem of 'indefinite integration' or of 'finding a function whose differential coefficient is a given function'. These descriptions are vague and in some ways misleading; and it is necessary to define our problem more precisely before we proceed further.
Let us suppose for the moment that is a real continuous function of the real variable . We wish to determine a function whose differential coefficient is , or to solve the equation A little reflection shows that this problem may be analysed into a number of parts.
We wish, first, to know whether such a function as necessarily exists, whether the equation (1) has always a solution; whether the solution, if it exists, is unique; and what relations hold between different solutions, if there are more than one. The answers to these questions are contained in that part of the theory of functions of a real variable which deals with 'definite integrals'. The definite integral which is defined as the limit of a certain sum, is a solution of the equation (1). Further where is an arbitrary constant, is also a solution, and all solutions of (1) are of the form (3).
These results we shall take for granted. The questions with which we shall be concerned are of a quite different character. They are questions as to the functional form of when is a function of some stated form. It is sometimes said that the problem of indefinite integration is that of 'finding an actual expression for when is given'. This statement is however still lacking in precision. The theory of definite integrals provides us not only with a proof of the existence of a solution, but also with an expression for it, an expression in the form of a limit. The problem of indefinite integration can be stated precisely only when we introduce sweeping restrictions as to the classes of functions and the modes of expression which we are considering.
Let us suppose that belongs to some special class of functions . Then we may ask whether is itself a member of , or can be expressed, according to some simple standard mode of expression, in terms of functions which are members of . To take a trivial example, we might suppose that is the class of polynomials with rational coefficients: the answer would then be that is in all cases itself a member of .
The range and difficulty of our problem will depend upon our choice of (1) a class of functions and (2) a standard 'mode of expression'. We shall, for the purposes of this tract, take to be the class of elementary functions, a class which will be defined precisely in the next section, and our mode of expression to be that of explicit expression in finite terms, i.e. by formulae which do not involve passages to a limit.
One or two more preliminary remarks are needed. The subject-matter of the tract forms a chapter in the 'integral calculus'[1], but does not depend in any way on any direct theory of integration. Such an equation as is to be regarded as merely another way of writing (1): the integral sign is used merely on grounds of technical convenience, and might be eliminated throughout without any substantial change in the argument.
The variable is in general supposed to be complex. But the tract should be intelligible to a reader who is not acquainted with the theory of analytic functions and who regards as real and the functions of which occur as real or complex functions of a real variable.
The functions with which we shall be dealing will always be such as are regular except for certain special values of . These values of we shall simply ignore. The meaning of such an equation as is in no way affected by the fact that and have infinities for .