EXPANSION PROBLEMS. 387 



must occur in conjugate pairs. -^^ Returning to the rejected part of 

 the sector (4), we need not inquire minutely what roots may be present 

 there ; it is sufficient for our purposes to recognize that, as <p is analytic 

 in the finite region in question, there can not be infinitely many of 

 them. There will be an index g, positive, negative, or zero, such that 

 if the characteristic numbers ^^ are arranged in order of increasing 

 absolute value, and the first is denoted by Pg, the second by Pg+i, 

 and so on, the number pn will be nearly equal to p'n, for large values of n. 

 To summarize : 



The system (1), (2) has infinitely many characteristic values on, of 

 which not more than a finite number can be imaginary. ^^ The 

 characteristic numbers can be written in the form 



Pn = —]=+ -i=-\-en, 11= g,g-\-l,g-\-2,..., 



3 V3 V.S 



where 



lun f. 



en = 0. 

 n= 00 



To this we may add at once that there is just one characteristic 

 function un{x) corresponding to each characteristic number, 



Un (x) = e-""^ + coe""""^ + oi'^c-'^P'^^ 



(7) f I- ^3 V3 \ 

 = 6-""^+ e J""-' ( V3 sin — PnX - cos — pnxj, 



the latter expression involving only real quantities if pn is real. 



Now if the usual expansion theory were capable of extension to the 

 present problem, it would be possible to develop any function, satisfy- 

 ing certain not very restrictive conditions, in a uniformly convergent 



series of the form 



00 



(8) 2 an Unix). 



n = g 



But we shall see that the series is very far from possessing the necessary 

 flexibility. In fact, we shall prove the following theorem : 



10 Cf . Bocher, Boundary problems in one dimension, International Congress 

 of Mathematicians, Cambridge 1912; see the last paragraph of §6, containing 

 a reference to Birkhofl. 



11 If some of the characteristic numbers in the finite region just mentioned 

 should lie on the rays that form the boundary of the sector, we should take 

 into account only the values on one of the rays. 



12 Laudien (loc. cit., pp. 73-76) gives a proof, which he ascribes to Teich- 

 mann, that the characteristic values are real from the very beginning. 



