114 BIRKHOFF AND LANGER. 



Proof: Writing X = pe t8 we have 



<P2(X) = <pi{pe x ) = \p 2 (e, p). 

 Hence 



1 f M A \ C 



, hm ^Z:J ^ 2 (X)-= lira - / M9,p)dd, 

 |x|=oo 1*1 Ca g X '=« 2ir e a 



and since the limit of the integral on the right is zero by lemma 1, 



lim ^J^, 2 (X)- = 0. Q.E.D. 



|x|-oo 2m Ca0 X 



Lemma 3 : Given any function <p 3 (x, X) which is such that 



(i) | <p 3 (x, X) | < M for a ^ .r ^ /?, a g argX ^ 0^, | X | > N> 



(104) 



(ii) lim tp 3 (x, X)| = uniformlv for a ^x^ /?— x, 

 |x|=» 



a ^argX ^ 00-0 x , 

 then /(X) = / <p s (x, X) rfx is a function of the type <p 2 (X) of lemma 2. 



a 



Proof: When X is confined to the sector d a ^ arg X ^ dp — 9 X , 



lim /(X) = uniformlv by lemma 1, while for X in the larger 

 |x|=» 



sector 6 a ^j arg X ^ 0^ 



I /(X) | ^ J M(& = M(/3 - a) . 



a 



Hence I(\) is of type <p 2 (X) by definition. Q. E. D. 



As a matter of convenience we shall use hereafter the symbols 

 <pi, <p2, and <p3, to designate any function of X and other arguments 

 which satisfies conditions similar to (102), (103) or (104) respectively. 

 As heretofore functions which approach the limit zero uniformly as 

 | X | increases indefinitely will be denoted by e. Let us return now to 

 the direct evaluation of S m (.r) • . 



