The term `theory of functions' was first used by Lagrange (Th4orie des fonctions analytigues. Paris, 1797). The branch thus denoted deals with functions of more general than those described above. E.g. in the equation w=f (z), must, in general, be taken to be a complex ber (q.v.), x + yi, where i stands for The theory, therefore, has for its object the study of functions of one or more variables, in 'which either the variables or the coefficients, or both, are complex numbers. This general theory may be said to have been founded by Lagrange (1772, 1797, 1806), although Newton, Leibnitz, Johann Bernoulli, Clairaut (1734), D'Alembert (1747), and Euler (1753) had already worked toward it. Gauss contributed to the theory, especially in its application to the fundamental theorem of algebra. Cauchy, starting from Lagrange's work, greatly developed it. and numerous propositions due to him arc found in the various text-books on the subject. His memoirs extend over a period of nearly forty years (1814.51), covering a large part of the general theory as known to-day, and ing the subject upon a firm foundation. The torical development after Cauchy's time becomes interwoven with that of special functions. notably the elliptic and Abelian.
Elliptic functions arose from the consideration of the integral where 12 is a rational ?X f (a.) , and X is the general rational and integral quartie + a„.z + a,. The theory of these functions had been suggested by .Jakob Bernoulli (1691) and by Maclaurin (1742). and D'Alembert (1746) had approached it. Euler had gone further (from 1761) and had prophe sied (1766) that there would come "a new sort of calculus of which 1 have here attempted the ex position of the first elements." To Landen (1775), however, the honor is usually given of founding the theory. But it is to Legendre that its real development is due. He worked forty years in perfecting it, his labor culminating in his Traite des functions elliptiques et des int6 grates Euleriennes (1825-28). At the same time
that Legendre published this work, Abel and Jacobi began their great contributions. Abel, whose fundamental theorem was not published until after his death, discovered the double periodicity of elliptic functions. Jacobi created a new notation and gave name to the 'modular equations' of which he made use. Cayley con tributed to the subject in England, his only book being devoted to it.
The general theory of functions has received its present form largely from the works of Cauchy, Riemann, and Weierstrass. Endeavoring to sub ject all natural laws to mathematical interpre tation, Riemann attacked the subject from the standpoint of the concrete, while Weierstrass pro ceeded from a purely analytic point of view. Riemann's theories have been elaborated by Clebsch, and also by Klein, who has materially extended the theory of Riemann's surfaces, and who has generalized Clebsch's application of mod ern geometry to the study of elliptic functions in his Thcorie der elliptischen Modulfunctionen. This last-named theory had its origin in a mem oir of Eisenstein (1847), and in the lectures of Weierstrass on elliptic functions.
In the theory of functions, the number of spe cial functions is very great. For the list at the present time, consult: Miller, "Mathematisehe Terminologie," in Bibliotheca Mathematica(Leip zig, 1901), where some two hundred are men tioned. The most notable work on the historic development of functions is that of Brill and Noether, "Die Entwickelung der Theorie der algebraischen Functionen in alterer und neuerer Zeit," in Jahresbericht der deutschen Mathe matiker Vereinigung, vol. ii. (Berlin, 1894). For theory, bibliography, and historical notes, con sult: Harkness and Morley, Theory of Functions (New York, 1893) ; and Forsyth, Theory of Func tions (Cambridge, 1893). For further bibliog raphy of historical development, and for articles on the theory of functions, consult Merriman and Woodward, Higher Mathematics (New York, 1896).