400 JACKSON. 



shall use. Or, rather, we shall use this form if the numbers 5i and 82, 

 which were introduced earlier and figure now in the coefficients of yi 

 and yo, are conjugate imaginaries, but shall multiply it by i if 5i is 

 the negative of the conjugate of 62. We have then in either case 



V 



(22) un (.r) = Z [(Jt] e'"'"''^ 



where di and do, in particular, are conjugate imaginary quantities 

 different from zero. It is clear that if x is positive and n is large, all 

 the terms after the second will be insignificant in comparison with 

 either of the first two. Let 



where Jh and I12 are real. Then 



(23) di e""""^ + di e""^^^ = 2e^' + p,.xcos{^/r) ^^g | f^^ _^ ^^^ gjj-^ (^/j,) | _ 



This identity is of course none the less valid for the circumstance that 

 the numbers pn are not necessarily real. 



Suppose that a succession of constants ^^ an, 71 = g, g -]r 1, g -^ 2, . . . , 

 is such that anUn(x) remains uniformly finite throughout an interval 

 included in (0, tt), or, in symbols, 



(24) I anUn (x) I < G 



for Xi < X < xo, where < .I'l < .I'o ^ tt. It is to be shown that 

 throughout the interval ^ .r < xo the series 



(25) Y. ttnUnix) 



n = g 



must converge and represent a function possessing derivatives of all 

 orders. 



Because of the relations (20) and (21), the difference between the 

 last factor in (23) and the expression 



(26) cos { h2 + (mo + n) x ] 



25 This notation is adequate only if all the characteristic values from the 

 very beginning are simple roots of the determinant equation. In order not to 

 raise questions irrelevant to the main issue, it will be assumed, in the contrary 

 case, that the series (25) begins with such a value of the index n that only 

 simple roots are involved. In the extension of Theorem II, where the formal 

 expansion of a definite function is under discussion, it is possible to assign 

 values to the first terms of the expansion immediately; cf. Birkhoff II, p. 380; 

 but we shall not be in any way directly concerned with these terms. 



