BOUNDARY PROBLEMS AND DEVELOPMENTS. 113 



where x is an arbitrarily small positive constant, then 



(8 



lim / <pi(x, X) dx = 



I — «■> t7 



|X|=oo 



a 



Proof: Under the hypotheses (102) we have 



P-x 



lim / ipi(.r, X) dx — uniformly, 



|X| = oo d 

 a 



and | J Vl (x, X) dx | < ilf x, 



/3-X 



while it follows from these relations and the relation 

 P-x 



I J <Pi(x, X) dx | g | J ^{x, X) tf.r | + | J <Pi(x, X) dx |, 



a a P-x 



that 





X) dx | < 23/ x 



for | X | sufficiently large. Since x is arbitrary, however, this means 

 that 



P 



lim / ^i(.T, X) <fc = 0. Q. E. D. 



|x|=«| ° 



a 



Lemma 2: Given any function <P'i(\) which is such that 



(i) | <p 2 (\) | < M for 6 a ^ arg X ^ O , | X | > N 



1 .... lim | <p 2 (X) | = uniformly, for# a ^ arg X ^ 0^ - b >t 



\ n ) |X|=oo 



where X is a constant arbitrarily small but positive, then 



p 2 (X) — = 0, 



lim J_ f - d * 



I x l= w 2™'* 



C Q/8 being that arc of the circle | X | = p which lies in the sector bounded 

 by arg X = 6 a and arg X = dp. 



