BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 



523 



points of C. These restric- 

 tions on a (.r) ensure that we 

 can choose regular curves Di 

 and D2, within and without 

 C respectively, and osculating 

 C at the points where a (.r) is 

 not analytic (Fig. 1) in such a 

 way that on the continua lim- 

 ited by Di, D2, we have 



(1) 



a (.r) I ^ K, 



a (x) — a (x) 



Fig. 1. 



^K; 



in these continua a (x) is defined as the analytic extension of a (x) on 

 C. It is possible to extend further the definition of a (.r) throughout 

 the ring formed by Ci, C2 in such wise that inequalities of the type 

 (1) hold; for this purpose it is clearly sufficient to choose real and 

 imaginary components that join on continuously to the like com- 

 ponents of a (x) along Di and Do, and to make each component satisfy 

 inequalities of the same nature as (1). Such a choice can always be 

 made. 



§ 2. On a First Type of Integral. 



Let us turn now to consider the integral 



(2) 



i_ frmiMam, 



2t^—1 Jc t~x 



where g'^(x) is a function analytic within C and continuous along C, 

 and p is zero or a positive integer. Following Plemelj (loc. cit.) we 

 shall term a function g+(x) of this description a regular inner function 

 and affix to it a superscript + ; likewise a superscript — will indicate 

 that a function is a regular outer function, i. e. is analytic in the 

 extended plane without C, and continuous along C. 



We can demonstrate at once that the integral (2) represents a regu- 

 lar inner function f^{x), or a regular outer function f~{x), according 

 as .T is within or without C. In the first place these functions are 

 analytic within and without C respectively, as appears from (2). 



In the second place, by Cauchy's integral theorem we have 



(3) 



2 7rV— l^C t — x 



it) 



di 



1^ 



2 7rV: 





(t)g^(t) 



dt, 



