MR. G. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. 



In particular, we notice that this theorem is true if 4> be an integral function. 

 The result of the last section concerning exp {/(*)} is, of course, a particular case of 

 the theorem. 



5. Theorem HI. Let f(x) possess an asymptotic expansion with constants and 

 oharaotorittict A, B, k, I, p, a-, valid when \x\^. y and for a certain range of values 

 of arg z. Then, if <J>, (a +) be a function of which is regular ivlien | f | S c, u-licn- 

 c<A/* + By~' (ft having its usucd signification), then, when \x\ is greater than 

 the two quantities y and B/c, 4> {/()} possesses an asymptotic expansion, valid 

 for the same range of values of arg x as the expansion for f(x), with characteristics 

 k, I, p, <r a , wlmre p a is the gi-eater of the quantities* p and p. (AX/c), while 



where y, is the larger of the quantities y and B/c. 

 Let the expansion of 4>, (a + f ), when | | S c be 



since this series is absolutely convergent when | | = c, we have 



I |A.|c"<H, | A. | < He-, 



vfl 



where H is finite and independent of m. 

 Since f(x) = a + E , we may expand <J>, {/()} into the series 



provided |R,jSc. 

 Since |R |s E\x\ ~\ the expansion will be valid when 



|*|iy and E\x\~ l Sc. 

 Substituting for R,, R, 8 ..... R,- ^ in Theorem II., we get 



...... (9) 



where 



rf. = /J , U = S A.R,- 

 and wlien n > 



*=?,* * U.= SA..A+ S A.R.-. 



m = m = n+1 



The quantities .6., .8. are the same as those which occur in equation (6). 



", Po = P + &, where S is an arbitrary positive quantity as small as we please. 



