A FEW EXAMPLES IN THE THEORY OF FUNCTIONS 24I 



In the expansion (7) the ratio of a term and the term preceding it is 



- — r , so that the expansion is convergent for every point z within a circle 



having b as a center and 1 1 — b | as a radius. Next, any point c within this 

 circle may be chosen and the series (7) be expanded about this point. 

 In this manner an expansion }(z/a/b/c) is obtained which converges 

 within a circle having c as a center and 1 1 — c | as a radius. As is indicated 

 in Fig. 2, this process of continuing the given function beyond its original 

 region of convergence may be carried on indefinitely, so that ultimately 

 the whole plane has been covered. All these continuations, together 



represent the analytic function. -— . 



4. Example III. 1 The many- valued function w determined by 



THE EQUATION W 3 — W + Z = 0. 



In the Elements of the Theory of Functions of a Complex Vari- 

 able by Durege (English translation) the function w of z denned by 

 the equation w 3 -w+z=o, as an example of a multiform function is 

 studied (p. 38 and p. 76). The treatment however does not seem to 

 be very clear and I shall therefore attempt to analyze the various parts 

 of the problem with the necessary details. 



(a) Algebraic Part. 

 Putting 



,^l(_ 2+A /,-A) , 



a=-(-I+*l/i) , 



2 



1 For similar and other examples see Harkness and Morley, A Treatise on the Theory of Functions, 

 pp. 127-150; Harkness and Morley, Introduction to the Theory of Analytic Functions, pp. 284-286; Picard 

 Traiti d'Analyse, Vol. II, pp. 348-388 (1 ed.); Appeix & Ooursat, Thiorie des Fonctions Algibriques; 

 Fricke, Analytisch-Funklionentheoretische Vorlesungen, pp. 75-172; Landfriedt, Theorie der algebraischen 

 Funklionen und ihrer Integrale; Bianchi, Lezioni sulla Teoria delle Funzioni di VariobiU Complessa, pp. 

 317-246. 



