The number of essentially different types of functions with which we have been concerned in the foregoing chapters is not very large, the most important being polynomials, rational functions, algebraical functions, explicit or implicit, and trigonometrical functions, direct or inverse.

The gradual expansion of mathematical knowledge has been accompanied by the introduction into analysis of one new class of function after another. These new functions have generally been introduced because it appeared that some problem which was occupying the attention of mathematicians was incapable of solution by means of the functions already known. The process may fairly be compared with that by which the irrational and complex numbers were first introduced, when it was found that certain algebraical equations could not be solved by means of the numbers already recognised. One of the most fruitful sources of new functions has been the problem of integration. Attempts have been made to integrate some function f(x) in terms of functions already known. These attempts have failed; and after a certain number of failures it has begun to appear probable that the problem is insoluble. Sometimes it has been proved that this is so; but as a rule such a strict proof has not been forthcoming until later on. Generally it has happened that mathematicians have taken the impossibihty for granted as soon as they have become reasonably convinced of it, and have introduced a new function F(x) defined by its possessing the required property, viz. that F′(x) = f(x).