390 JACKSON. 



If a function /(.c) is the sum of a uniformly convergent series (8), 

 the coefficients in the series must be the numbers 



(13) On = —^ 



A Un{x)Vn{x)dx 



The number N being given, let q be the first number greater than or 

 equal to N which is congruent to 2 (mod. 3). We shall examine the 

 terms of the formal series for the special function 



.T5+l(7r-x)«+2 



(14) fix) 



TT^ + ^X^+l)! 



This function, a polynomial in x, is of course analytic from to tt, 

 and vanishes with its first q derivatives at both ends of the interval, 

 while the (q + l)th derivative takes on the value at the point tt 

 and the value 1 at the point 0. 



To determine the order of magnitude of the numerator in (13), 

 when f{x) is the function just defined, let us break up the integral 

 into three, corresponding to the three terms in the right-hand member 

 of (12), and consider for the moment the second of these. The index 

 n may be supposed so large that pn is real and positive. Integrating 

 by parts q -\- 2 times, we find that 



J"' /—IN «+2 

 / (.t) e"""^''—^^ dx = g-^Pn-T 



\ wpn J 



+ f ^1 '^^ fV ^'+^^ (^) e """^^-'^ dx, 



\ copn J J 



and hence, by one more integration by parts and the application of 

 obvious inequalities, that 



L 



f (.r) e'^''''(^-'^) dx — g- a,p„^ 



' \wpnj 



< 



Pn 



g+3 



c 



:= p ^prni 



Pn^+-' ' 



where c is independent of n A corresponding inequality holds for 

 the third integral, with the same value of c. As for the first, it ob- 

 viously remains finite ^^ as n becomes infinite, and so it is certainly 

 true that 



Jo 



/ (x) gP''^^-'^) dx I < — x^ e*"'"' 



I Pn^'^'^ 



16 As a matter of fact, of course, it approaches zero. 



