VI. Transcendental functions

1. The theory of the integration of transcendental functions is naturally much less complete than that of the integration of rational or even of algebraical functions. It is obvious from the nature of the case that this must be so, as there is no general theorem concerning transcendental functions which in any way corresponds to the theorem that any algebraical combination of algebraical functions may be regarded as a simple algebraical function, the root of an equation of a simple standard type.

It is indeed almost true to say that there is no general theory, or that the theory reduces to an enumeration of the few cases in which the integral may be transformed by an appropriate substitution into an integral of a rational or algebraical function. These few cases are however of great importance in applications.

2. (i) The integral where is an algebraical function, and commensurable numbers, can always be reduced to that of an algebraical function. In particular the integral where is rational, is always an elementary function. In the first place a substitution of the type will reduce it to the form and then the substitution will reduce this integral to the integral of a rational function.

In particular, since and are rational functions of , and and are rational functions of , the integrals} are always elementary functions. In the second place the substitution just indicated is imaginary, and it is generally more convenient to use the substitution which reduces the integral to that of a rational function, since

(ii) The integrals are included in the two standard integrals above.

Let us consider some further developments concerning the integral[48] If we make the substitution , the subject of integration becomes a rational function , which we may suppose split up into

(a) a constant and certain positive and negative powers of ,

(b) groups of terms of the type

The terms (i), when expressed in terms of , give rise to a term In the group (1) we put , and, using the equation we obtain a polynomial of degree in . Since this polynomial may be transformed into the form

The function is now expressed as a sum of a number of terms each of which is immediately integrable. The integral is a rational function of and if all the constants vanish; otherwise it includes a number of terms of the type

Let us suppose for simplicity that , when split up into partial fractions, contains no terms of the types Then and the constants may be determined by multiplying each side of the equation by and making tend to .

It is often convenient to use the equation which enables us to decompose the function into two parts and such that If has the period , then must vanish identically; if it changes sign when is increased by , then must vanish identically. Thus we find without difficulty that, if , or where , according as is odd or even.

Similarly

(iii) One of the most important integrals in applications is where and are real. This integral may be evaluated in the manner explained above, or by the transformation . A more elegant method is the following. If , we suppose positive, and use the transformation which leads to If , we suppose positive, and use the transformation

The integral may be reduced to this form by the substitution , where . The forms of the integrals may be deduced by the use of formulae of reduction, or by differentiation with respect to . The integral is really of the same type, since And similar methods may be applied to the corresponding integrals which contain hyperbolic functions, so that this type includes a large variety of integrals of common occurrence.

(iv) The same substitutions may of course be used when the subject of integration is an irrational function of and , though sometimes it is better to use the substitutions , , or . Thus the integral where is reduced to an elliptic integral by the substitution . The most important integrals of this type are

3. The integral where are any numbers (commensurable or not), and is a polynomial, is always an elementary function. For it is obvious that the integral can be reduced to the sum of a finite number of integrals of the type and This type of integral includes a large variety of integrals, such as (, , , being positive integers) for which formulae of reduction are given in text-books on the integral calculus.

Such integrals as where is a polynomial, may be reduced to particular cases of the above general integral by the obvious substitutions

4. Except for the two classes of functions considered in the three preceding paragraphs, there are no really general classes of transcendental functions which we can always integrate in finite terms, although of course there are innumerable particular forms which may be integrated by particular devices. There are however many classes of such integrals for which a systematic reduction theory may be given, analogous to the reduction theory for elliptic integrals. Such a reduction theory endeavours in each case

(i) to split up any integral of the class under consideration into the sum of a number of parts of which some are elementary and the others not;

(ii) to reduce the number of the latter terms to the least possible;

(iii) to prove that these terms are incapable of further reduction, and are genuinely new and independent transcendents.

As an example of this process we shall consider the integral where is a rational function of .[49] The theory of partial fractions enables us to decompose this integral into the sum of a number of terms

Since the integral may be further reduced so as to contain only

(i) a term where is a rational function;

(ii) a number of terms of the type If all the constants vanish, then the integral can be calculated in the finite form . If they do not we can at any rate assert that the integral cannot be calculated in this form[50]. For no such relation as where is rational, can hold for all values of . To see this it is only necessary to put and to expand in ascending powers of . Then and no logarithm can occur in any of the other terms[51].

Consider, for example, the integral This is equal to and since and we obtain finally Similarly it will be found that this integral being an elementary function.

Since if , all integrals of this kind may be made to depend on known functions and on the single transcendent which is usually denoted by and is of great importance in the theory of numbers. The question of course arises as to whether this integral is not itself an elementary function.

Now Liouville[52] has proved the following theorem: 'if is any algebraical function of , and is an elementary function, then being rational functions of and the degree of the algebraical equation which determines as a function of '.

Liouville's proof rests on the same general principles as do those of the corresponding theorems concerning the integral . It will be observed that no logarithmic terms can occur, and that the theorem is therefore very similar to that which holds for in the simple case in which the integral is algebraical. The argument which shows that no logarithmic terms occur is substantially the same as that which shows that, when they occur in the integral of an algebraical function, they must occur linearly. In this case the occurrence of the exponential factor precludes even this possibility, since differentiation will not eliminate logarithms when they occur in the form

In particular, if is a rational function, then the integral must be of the form and this we have already seen to be impossible. Hence the 'logarithm-integral' is really a new transcendent, which cannot be expressed in finite terms by means of elementary functions; and the same is true of all integrals of the type which cannot be calculated in finite terms by means of the process of reduction sketched above.

The integrals may be treated in a similar manner. Either the integral is of the form or it consists of a term of this kind together with a number of terms which involve the transcendents which are called the cosine-integral and sine-integral of , and denoted by and . These transcendents are of course not fundamentally distinct from the logarithm-integral.

5. Liouville has gone further and shown that it is always possible to determine whether the integral where are algebraical functions, is an elementary function, and to obtain the integral in case it is one[53]. The most general theorem which has been proved in this region of mathematics, and which is also due to Liouville, is the following.

'If are functions of whose differential coefficients are algebraical functions of , and denotes an algebraical function, and if is an elementary function, then it is of the form where are algebraical functions of . If the differential coefficients are rational in , and is rational, then are rational in .'

Thus for example the theorem applies to since, if the various arguments of are denoted by , , , , , , , we have The proof of the theorem does not involve ideas different in principle from those which have been employed continually throughout the preceding pages.

6. As a final example of the manner in which these ideas may be applied, we shall consider the following question:

'in what circumstances is where is rational, an elementary function?'

In the first place the integral must be of the form A general consideration of the form of the differential coefficient of this expression, in which must only occur linearly and multiplied by a rational function, leads us to anticipate that (i) must be of the form where , , and are rational, and (ii) must be rational functions of only; so that the integral can be expressed in the form

Differentiating, and comparing the result with the subject of integration, we obtain the equations Hence is a constant, say , and

We can always determine by means of elementary operations, as in iv., §4, whether this integral is rational for any value of or not. If not, then the given integral is not an elementary function. If is rational, then we must calculate its value, and substitute it in the integral which must be rational for some value of the arbitrary constant implied in . We can calculate the rational part of the transcendental part must be cancelled by the logarithmic terms

The necessary and sufficient condition that the original integral should be an elementary function is therefore that should be of the form where is a constant and is rational. That the integral is in this case such a function becomes obvious if we integrate by parts, for

In particular are not elementary functions unless in (i) and in (ii) . If the integral is elementary then the integration can always be carried out, with the same reservation as was necessary in the case of rational functions.

It is evident that the problem considered in this paragraph is but one of a whole class of similar problems. The reader will find it instructive to formulate and consider such problems for himself.

7. It will be obvious by now that the number of classes of transcendental functions whose integrals are always elementary is very small, and that such integrals as where is algebraical, or even rational, are generally new transcendents. These new transcendents, like the transcendents (such as the elliptic integrals) which arise from the integration of algebraical functions, are in many cases of great interest and importance. They may often be expressed by means of infinite series or definite integrals, or their properties may be studied by means of the integral expressions which define them. The very fact that such a function is not an elementary function in so far enhances its importance. And when such functions have been introduced into analysis new problems of integration arise in connection with them. We may enquire, for example, under what circumstances an elliptic integral or elliptic function, or a combination of such functions with elementary functions, can be integrated in finite terms by means of elementary and elliptic functions. But before we can be in a position to restate the fundamental problem of the Integral Calculus in any such more general form, it is essential that we should have disposed of the particular problem formulated in Section III.

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