200 MR. O. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. 



NO that 



Now, by the asymptotic expansion of the gamma function, it follows that 



for all integer values of n, where K is finite and independent of n, so thai/ 

 |af +1 T.|<[y{exp(A/ t +By- 1 )-exp(A/ i )}+Kexp(By- 1 )] x |expa |.r(/+l)o- B . (7u) 



Combining the equations (7 A) and (7fi), we see from (7) that exp f(x) possesses an 

 asymptotic expansion of which k, I, p, cr are characteristics. 



4. We shall conclude this part of the paper by proving three general theorems. 



Theorem II. Let f(x) possess an asymptotic expansion with characteristics 

 k, I, p, a; and constants A, B. Then, if < (f ) be a function of g ivhich is regular* 

 inside a circle of radius greater than A/A + By" 1 , the centre of the circle being at 

 = a (/i and y having their usual significations], then <fr {f(x)} possesses an 

 asymptotic expansion with characteristics k, I, p, a; ivhich is valid for the same range 

 of values of x as the expansion off(x). 



Let "^(ao+f) be expansible in the series 



Then 



and hence 



where G is finite and independent of m. [These statements are true if the less 

 stringent condition be satisfied.] 

 Now, f(x) = a +Ko, so that 



this series being convergent since |R| < By" 1 . 



u condition may be replaced by the slightly less stringent condition that the series for *(oo + ) 

 should be absolutely convergent when || = A/* + By -1 . 



