MATHEMATICS IN THE NINETEENTH CENTURY 475 



for example, existence theorems for implicit functions and the solu- 

 tions of certain differential equations, the development of functions 

 in infinite series and products, and the periods of integrals of one 

 and many valued functions. 



Meanwhile Germany is not idle; Weierstrass and Riemann de- 

 velop Cauchy's theory along two distinct and original paths. Weier- 

 strass starts with an explicit analytical expression, a power series, 

 and defines his function as the totality of its analytical continua- 

 tions. No appeal is made to geometric intuition, his entire theory 

 is strictly arithmetical. Riemann growing up under Gauss and 

 Dirichlet not only relies largely on geometric intuition, but he also 

 does not hesitate to impress mathematical physics into his service. 

 Two noteworthy features of his theory are the many leaved surfaces 

 named after him, and the extensive use of conformal representation. 



The history of functions as first developed is largely a theory of 

 algebraic functions and their integrals. A general theory of func- 

 tions is only slowly evolved. For a long time the methods of Cauchy, 

 Riemann, and Weierstrass were cultivated along distinct lines by 

 their respective pupils. The schools of Cauchy and Riemann were 

 the first to coalesce. The entire rigor which has recently been im- 

 parted to their methods has removed all reason for founding, as 

 W T eierstrass and his school have urged, the theory of functions on 

 a single algorithm, namely, the power series. We may therefore say 

 that at the close of the century there is only one theory of functions 

 in which the ideas of its three great creators are harmoniously united. 



Let us note briefly some of its lines of advance. Weierstrass early 

 observed that an analytic expression might represent different 

 analytic functions in different regions. Associated with this is the 

 phenomenon of natural boundaries. The question therefore arose, 

 What is the most general domain of definition of an analytic function? 

 Runge has shown that any connected region may serve this purpose. 

 An important line of investigation relates to the analytic expression 

 of a function by means of infinite series, products, and fractions. 

 Here may be mentioned Weierstrass's discovery of prime factors; 

 the theorems of Mittag-Leffler and Hilbert; Poincare's uniform- 

 ization of algebraic and analytic functions by means of a third 

 variable, and the work of Stieljes, Fade, and Van Vleck on infinite 

 fractions. Since an analytic function is determined by a single 

 power series, which in general has a finite circle of convergence, two 

 problems present themselves: determine, first, the singular points of 

 the analytic function so defined, and, second, an analytic expression 

 valid for its whole domain of definition. The celebrated memoir of 

 Hadamard inaugurated a long series of investigations on the first 

 problem; while Mittag-Leffler 's star theorem is the most important 

 result yet obtained relating to the second. 



