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



In like manner, 8 ince f V* is convergent, we may show that, when n is any 



finite integer, 



&- F (?/) = I ff M (./) e~ } ' dt when y == 0, 

 cfy" Jo 



and /* denotes the **" differential coefficient of/ 

 Putting y = 0, we get 



= /<"> (0) rt- d, ^e., / (B) (0) = 0. 



Therefore /(y) is analytic when | ? y| < 2 A and all the differential coefficients of f(y) 

 vanish when y = ; that is to say, /(y) is a pure constant, which we wil 

 Furthermore, by the definition of f(y) 



L< Aexp(-|y|) 



when |argy|Ssfir+X, for all values of |y| , no matter how large ; since exp (- \y\ )* 

 as jy|*- 0. we infer that L = 0. 



We have thus proved the lemma, that 



8. We are now in a position to discuss the uniqueness of an analytic function 

 possessing an asymptotic expansion with given characteristics for a certain range of 

 values of the argument of the variable. 



The theorem, stated precisely, is as follows : 



Theorem V.Let there be two functions /, (?;), / 2 (x), which are analytic in the 

 region dejined by the inequalities 



and let them be such that in this region they possess the asymptotic expansions 



x 



where, for all values ofn, 



Then, ?//8->ir/, 



Let the i-egiou in which x is permitted to lie be called the region C. 

 Since 



