Theory of Functions

Abstract

IF evidence were wanted of the recent progress of the study of pure mathematics on English and American soil, none better could be furnished than the appearance on the two sides of the Atlantic, within a short interval, of two important works on the theory of functions of a complex variable. But a few years ago this great modern branch of mathematics was so little known to English-speaking mathematicians that scarcely a trace of its influence could be traced in their writings, and the majority of our text-books were disfigured by incompleteness, and not seldom by positive error arising from ignorance of its principles. Now the English reader has at his disposal two extensive works dealing with the fundamental principles of the theory from all the more important points of view; and also a very useful aid in Cathcart's valuable translation of Harnack's “Elements of the Differential and Integral Calculus.” Probably nothing could serve better as an exorcist of the spirit of formalism which has oppressed the English school of mathematicians so heavily, in spite of all the great things that its leaders have done for the science, than the study of the theory of functions. In no other mathematical discipline is the fundamental unity of logic kept so constantly before the student; nowhere else in mathematics is it so clearly made evident that the manifold array of symbols is the clothing, and not the soul of mathematical thought; and nowhere else can we perceive so fully that progress is to be looked for mainly in strengthening our hold upon elementary conceptions, in continual refinement of definition and continual increase of stringency in inference, together with the necessary complement of this, viz. a continual widening of our power of imagining logical possibilities.¹ A single illustration of these general remarks may be cited here, viz. the important part now played in mathematics by the classification of the possible singularities of a function. Although as yet this classification has hardly proceeded beyond the first stage of distinguishing between what Weierstrass has called essential and non-essential singularities, yet the exceeding fruitfulness of the idea is very manifest in every part, not only of the theory itself, but of its applications. In this connection we may remark that anyone who is sceptical as to the value of function-theory, should compare the treatment of the theory of elliptic functions as given in chapter vii. of the treatise now before us, with the older method of dealing with the same subject. He will there find the theorems which used to be for many of us a mere savagery of riotous mathematical formulæ, sitting now clothed in their right minds-the cultured dependents of a few leading ideas.

A Treatise on the Theory of Functions.

By Harkness Morley, Professor of Pure Mathematics in Haverford College, Pa. (London and New York: Macmillan and Co., 1893.)