Numbers of the Exponential Form. 47 



Borel and Fabry now says that, when the sequence 

 «oj «] a, 



is a " serie ecrite an hasard," — that is to say, a sequence 

 whose general term cannot be given in the manner explained 

 above, — this circle of convergence i> a line entirely composed 

 of essential singularities, so that (2) cannot be continued 

 beyond this line, ami represents the whole function. 



4. 



J have regarded the conception of an "aggregate of 

 definition" as an essential part of the cardinal and ordinal 

 theories of functions. By an ''aggregate of definition" 

 I understand any aggregate of values among those of the 

 independent variable such that, when the values of the (one- 

 valued) function are given for the points of merely this 

 aggregate, the values for all other points in the domain of 

 existence are determined. "When the domain of the variable 

 is the continuum of real numbers, the cardinal number of 

 this aggregate, when the function is continuous or analytic, is 



and the ordinal types are respectively 

 ?; and co or *co. 



Since, however, a knowledge of the means whereby the 

 value of the function at one point is calculated from its values 

 at other points (which varies for different classes of functions) 

 appears indispensable in addition to a knowledge of the 

 aggregate of definition, and the latter knowledge then follows 

 from the former, it might appear that the " aggregate of 

 definition ,J is always a superfluous conception. The following 

 example will show that this is not the case. 



Tt has long been known that a one-valued analytic function 

 f(z) reduces to a constant if it has a period smaller in absolute 

 amount than any assignable positive number. This pro- 

 position, without the necessary restriction to one-, or at least 

 iinite-valuedness. was treated, without complete justification, 

 as obvious by Jaeobi ; so that Weber in the reprint*, 

 edited by him, of Jacobi's memoir, has given a proof of 

 the proposition in question together with an analogous theorem 

 on one-vaiued continuous functions of a real variable. 



* " Ueber die vierfach periodischen Functionen zweier Variabeln, auf 

 die Bieh die Theorie deT AJbel'schen Transcendental! ttiitzt," von ( '. (i. .1. 

 Jaeobi (Crelle'a Journal. Bd. xiii. ( l v -"54) ; Ostwald's Klassikerd. exakten 

 Win. No. 64, hrsgb. von II. Weber, Leipzig, L895, pp. ■>>> :'>'■>). 



