III. The integration of elementary functions. Summary of results
Reading edition
In the following pages we shall be concerned exclusively with the problem of the integration of elementary functions. We shall endeavour to give as complete an account as the space at our disposal permits of the progress which has been made by mathematicians towards the solution of the two following problems:
(i) if is an elementary function, how can we determine whether its integral is also an elementary function?
(ii) if the integral is an elementary function, how can we find it?
It would be unreasonable to expect complete answers to these questions. But sufficient has been done to give us a tolerably complete insight into the nature of the answers, and to ensure that it shall not be difficult to find the complete answers in any particular case which is at all likely to occur in elementary analysis or in its applications.
It will probably be well for us at this point to summarise the principal results which have been obtained.
1. The integral of a rational function (iv.) is always an elementary function. It is either rational or the sum of a rational function and of a finite number of constant multiples of logarithms of rational functions (iv., 1).
If certain constants which are the roots of an algebraical equation are treated as known then the form of the integral can always be determined completely. But as the roots of such equations are not in general capable of explicit expression in finite terms, it is not in general possible to express the integral in an absolutely explicit form (iv.; 2, 3).
We can always determine, by means of a finite number of the elementary operations of addition, multiplication, and division, whether the integral is rational or not. If it is rational, we can determine it completely by means of such operations; if not, we can determine its rational part (iv.; 4, 5).
The solution of the problem in the case of rational functions may therefore be said to be complete; for the difficulty with regard to the explicit solution of algebraical equations is one not of inadequate knowledge but of proved impossibility (iv., 6).
2. The integral of an algebraical function (v.), explicit or implicit, may or may not be elementary.
If is an algebraical function of then the integral , or, more generally, the integral where denotes a rational function, is, if an elementary function, either algebraical or the sum of an algebraical function and of a finite number of constant multiples of logarithms of algebraical functions. All algebraical functions which occur in the integral are rational functions of and (v.; 11—14, 18).
These theorems give a precise statement of a general principle enunciated by Laplace[7]: 'l'intégrale d'une fonction différentielle (algébrique) ne peut contenir d'autres quantités radicaux que celles qui entrent dans cette fonction}'; and, we may add, cannot contain exponentials at all. Thus it is impossible that should contain or : the appearance of these functions in the integral could only be apparent, and they could be eliminated before differentiation. Laplace's principle really rests on the fact, of which it is easy enough to convince oneself by a little reflection and the consideration of a few particular cases (though to give a rigorous proof is of course quite another matter), that differentiation will not eliminate exponentials or algebraical irrationalities. Nor, we may add, will it eliminate logarithms except when they occur in the simple form where is a constant, and this is why logarithms can only occur in this form in the integrals of rational or algebraical functions.
We have thus a general knowledge of the form of the integral of an algebraical function , when it is itself an elementary function. Whether this is so or not of course depends on the nature of the equation which defines . If this equation, when interpreted as that of a curve in the plane , represents a unicursal curve, i.e. a curve which has the maximum number of double points possible for a curve of its degree, or whose deficiency is zero, then and can be expressed simultaneously as rational functions of a third variable , and the integral can be reduced by a substitution to that of a rational function (v.; 2, 7—9). In this case, therefore, the integral is always an elementary function. But this condition, though sufficient, is not necessary. It is in general true that, when is not unicursal, the integral is not an elementary function but a new transcendent; and we are able to classify these transcendents according to the deficiency of the curve. If, for example, the deficiency is unity, then the integral is in general a transcendent of the kind known as elliptic integrals, whose characteristic is that they can be transformed into integrals containing no other irrationality than the square root of a polynomial of the third or fourth degree (v., 20). But there are infinitely many cases in which the integral can be expressed by algebraical functions and logarithms. Similarly there are infinitely many cases in which integrals associated with curves whose deficiency is greater than unity are in reality reducible to elliptic integrals. Such abnormal cases have formed the subject of many exceedingly interesting researches, but no general method has been devised by which we can always tell, after a finite series of operations, whether any given integral is really elementary, or elliptic, or belongs to a higher order of transcendents.
When is unicursal we can carry out the integration completely in exactly the same sense as in the case of rational functions. In particular, if the integral is algebraical then it can be found by means of elementary operations which are always practicable. And it has been shown, more generally, that we can always determine by means of such operations whether the integral of any given algebraical function is algebraical or not, and evaluate the integral when it is algebraical. And although the general problem of determining whether any given integral is an elementary function, and calculating it if it is one, has not been solved, the solution in the particular case in which the deficiency of the curve is unity is as complete as it is reasonable to expect any possible solution to be.
3. The theory of the integration of transcendental functions (vi.) is naturally much less complete, and the number of classes of such functions for which general methods of integration exist is very small. These few classes are, however, of extreme importance in applications (vi.; 2, 3).
There is a general theorem concerning the form of an integral of a transcendental function, when it is itself an elementary function, which is quite analogous to those already stated for rational and algebraical functions. The general statement of this theorem will be found in vi., §5; it shows, for instance, that the integral of a rational function of , and is either a rational function of those functions or the sum of such a rational function and of a finite number of constant multiples of logarithms of similar functions. From this general theorem may be deduced a number of more precise results concerning integrals of more special forms, such as where is an algebraical function of (vi.; 4, 6).