II. Elementary functions and their classification
Reading edition
An elementary function is a member of the class of functions which comprises
(i) rational functions,
(ii) algebraical functions, explicit or implicit,
(iii) the exponential function ,
(iv) the logarithmic function ,
(v) all functions which can be defined by means of any finite combination of the symbols proper to the preceding four classes of functions.
A few remarks and examples may help to elucidate this definition.
1. A rational function is a function defined by means of any finite combination of the elementary operations of addition, multiplication, and division, operating on the variable .
It is shown in elementary algebra that any rational function of may be expressed in the form where and are positive integers, the 's and 's are constants, and the numerator and denominator have no common factor. We shall adopt this expression as the standard form of a rational function. It is hardly necessary to remark that it is in no way involved in the definition of a rational function that these constants should be rational or algebraical[2] or real numbers. Thus is a rational function.
2. An explicit algebraical function is a function defined by means of any finite combination of the four elementary operations and any finite number of operations of root extraction. Thus are explicit algebraical functions. And so is (i.e. ) for any integral values of and . On the other hand are not algebraical functions at all, but transcendental functions, as irrational or complex powers are defined by the aid of exponentials and logarithms.
Any explicit algebraical function of satisfies an equation whose coefficients are polynomials in . Thus, for example, the function satisfies the equation The converse is not true, since it has been proved that in general equations of degree higher than the fourth have no roots which are explicit algebraical functions of their coefficients. A simple example is given by the equation We are thus led to consider a more general class of functions, implicit algebraical functions, which includes the class of explicit algebraical functions.
3. An algebraical function of is a function which satisfies an equation whose coefficients are polynomials in .
Let us denote by a polynomial such as occurs on the left-hand side of (1). Then there are two possibilities as regards any particular polynomial . Either it is possible to express as the product of two polynomials of the same type, neither of which is a mere constant, or it is not. In the first case is said to be reducible, in the second irreducible. Thus is reducible, while both and are irreducible.
The equation (1) is said to be reducible or irreducible according as its left-hand side is reducible or irreducible. A reducible equation can always be replaced by the logical alternative of a number of irreducible equations. Reducible equations are therefore of subsidiary importance only; and we shall always suppose that the equation (1) is irreducible.
An algebraical function of is regular except at a finite number of points which are poles or branch points of the function. Let be any closed simply connected domain in the plane of which does not include any branch point. Then there are and only distinct functions which are one-valued in and satisfy the equation (1). These functions will be called the roots of (1) in . Thus if we write where , then the roots of in the domain are and , where
The relations which hold between the different roots of (1) are of the greatest importance in the theory of functions[3]. For our present purposes we require only the two which follow.
(i) Any symmetric polynomial in the roots of (1) is a rational function of .
(ii) Any symmetric polynomial in is a polynomial in with coefficients which are rational functions of .
The first proposition follows directly from the equations To prove the second we observe that so that the theorem is true for if it is true for . It is certainly true for It is therefore true for , and so for any symmetric polynomial in .
4. Elementary functions which are not rational or algebraical are called elementary transcendental functions or elementary transcendents. They include all the remaining functions which are of ordinary occurrence in elementary analysis.
The trigonometrical (or circular) and hyperbolic functions, direct and inverse, may all be expressed in terms of exponential or logarithmic functions by means of the ordinary formulae of elementary trigonometry. Thus, for example,
There was therefore no need to specify them particularly in our definition.
The elementary transcendents have been further classified in a manner first indicated by Liouville[4]. According to him a function is a transcendent of the first order if the signs of exponentiation or of the taking of logarithms which occur in the formula which defines it apply only to rational or algebraical functions. For example are of the first order; and so is where is defined by the equation and so is the function defined by the equation
An elementary transcendent of the second order is one defined by a formula in which the exponentiations and takings of logarithms are applied to rational or algebraical functions or to transcendents of the first order. This class of functions includes many of great interest and importance, of which the simplest are It also includes irrational and complex powers of , since, e.g., the function and the logarithms of the circular functions.
It is of course presupposed in the definition of a transcendent of the second kind that the function in question is incapable of expression as one of the first kind or as a rational or algebraical function. The function where is rational, is not a transcendent of the second kind, since it can be expressed in the simpler form .
It is obvious that we can in this way proceed to define transcendents of the th order for all values of . Thus are of the third, fourth, ...... orders.
Of course a similar classification of algebraical functions can be and has been made. Thus we may say that are algebraical functions of the first, second, third, ...... orders. But the fact that there is a general theory of algebraical equations and therefore of implicit algebraical functions has deprived this classification of most of its importance. There is no such general theory of elementary transcendental equations[5], and therefore we shall not rank as 'elementary' functions defined by transcendental equations such as but incapable (as Liouville has shown that in this case is incapable) of explicit expression in finite terms.
5. The preceding analysis of elementary transcendental functions rests on the following theorems:
(a) is not an algebraical function of ;
(b) is not an algebraical function of ;
(c) is not expressible in finite terms by means of signs of exponentiation and of algebraical operations, explicit or implicit[6];
(d) transcendental functions of the first, second, third, ... orders actually exist.
A proof of the first two theorems will be given later, but limitations of space will prevent us from giving detailed proofs of the third and fourth. Liouville has given interesting extensions of some of these theorems: he has proved, for example, that no equation of the form where are algebraical functions of , and different constants, can hold for all values of .