EXPANSION PROBLEMS. 



413 



When n is large, a characteristic solution may be obtained by setting 

 p = pn in the determinant Ai, and replacing the t-th element of the 

 last row by ijt{x, pn) = e""^''' [1], t = 1,2,. . ., v; and this is except for 

 a constant factor the only characteristic solution ^^ for p = pn. The 

 numbers pnWi are situated along the positive axis of reals, the numbers 

 pmvo and pniv^ along the rays making angles of 2ir/v with the positive 

 axis of reals, and so on. Hence, when the determinant is expanded 

 according to the minors of its last two rows, two terms are predominant 

 over the others when n is large and x is not equal to or tt, namely 

 the terms involving" gP"(«'iT+t^:x) ^^^ ^pn{mir+w,x) ^ ^pj^g coefficients 



of these expressions are quantities which have already been seen not 

 to vanish for large values of p. The limits approached by these 

 coefficients are equal in absolute value, and if the whole determinant 

 is divided by either square root of their product, they will be replaced 

 by limits which are conjugate imaginaries. Dividing out also the 

 factor e"""'"^, we may write 



(43) Un (x) = [di] e""^^^ + [do] e"""'^^, 



where f/i and do are conjugate imaginary quantities different from 

 zero,^^ and the limits indicated by the bracket symbols are approached 

 uniformly in x for .r' ^ x ^. x", if < x' < x" < tt. 



This expression for un{x) is sufficient to suggest the proof of the 

 first theorem, that a series of the form ^anun{x) which converges uni- 

 formly throughout an interval ^ .r < .tq must represent a function 

 continuous there with its derivatives of all orders. In calculating the 

 order of magnitude of the terms of the derived series it will be necessary 

 to go behind the relation (43) to the complete determinant expression 



venient. If the problem is real at the start, all the characteristic numbers 

 from a certain point on will be actually on the ray arg p = tt/v, and not merely 

 approach it asymptotically. For if p is a characteristic number, the conjugate 

 of p is a characteristic number, and e^^^ /" times the conjugate of p is a character- 

 istic number, and unless arg p = tv/p the number last obtained is distinct from 

 p, and there are two characteristic values where, if p is large, there ought to 

 be only one. It may be remarked that in the plane of the variable \ = p", 

 the characteristic values in this case are real from a certain point on, and 

 become negatively infinite. 



36 Cf . footnote in connection with the corresponding passage in the preceding 

 section. It is readily shown that the solution indicated can not vanish identi- 

 cally. 



37 When X = a number of terms are of the same order of magnitude, and 

 when X = TT the terms involving gp-Cwx+w^^) ^^^ ^Pn(w,x+W3^) become com- 

 parable with those indicated above. 



38 They are not the same as the numbers di and di of the preceding section. 



