EXPANSION PKOBLEMS. 395 



The last statement is still true if the sector Si is replaced by another 

 one, bounded by lines parallel to the sides of Si, and including the 

 latter sector wholly in its interior.^" Cutting off a part of the new 

 sector, if necessary, by a large circle about the origin, we will let Ti 

 stand for the remaining portion, throughout which each function 

 Vs {^> p)> with its derivatives, is analytic. Of course the regions Ti 

 will overlap each other to some extent, and where they do, two differ- 

 ent fundamental systems of solutions will have been defined; but as 

 we shall not have occasion to use the solutions in different regions Ti 

 simultaneously, no confusion will result. 



As the problem involves p only in the I'th power, attention may be 

 restricted to one fth of the p-plane, consisting, for example, of the 

 sectors Si, and iS2;'-i; if there should be characteristic values on the 

 boundary lines for which 



argp = ± -J 



V 



the values on one of these lines are to be disregarded. We have to deal, 

 then, only with the regions To and Top-i, and a third region U which 

 is finite in extent. It follows at once from the general existence 

 theorem that in this finite region a fundamental system of solutions 

 of the differential equation, and so the determinant whose vanishing 

 gives the characteristic values, may be taken as analytic in p. The 

 region can contain only a finite number of characteristic values, and 

 these need not concern us further. 

 Let us consider the region To. Let 



101 = 6", W2 = e" , 



while W3, . . ., wp, denote the remaining I'th roots of —1, arranged, for 

 example, in cyclic order, though the convention as to the last v — 2 

 subscripts is quite inessential.^^ Geometrically, the requirement is 

 that when the roots are represented in the complex plane, as the 

 vertices of a regular polygon of v sides inscribed in the unit circle, wath 



20 Birkhoff III, p. 117. The assertion is readily justified by replacing the 

 parameter p, in the argument of his paper I, by p + r, where r is a constant. 



21 Our use of subscripts here is different from Birkhoff 's; and in its next 

 stages our discussion, though following his in general method, is vitally affected 

 by the special nature of the boundary conditions. 



