298 MR d. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. 



Suppose that ;/ starts from y and describes a circle of radius \y a \ and centre at 

 the origin, ending at the point y^*". 



We have i?/ \ fi t \ < 7. 



F (.Vo)= /(tye)e-**(fc 



.'<}- JA) 



Making the point y move from y u to y e"' round the circle, we get 



F(^")= f 



J*r- 



Now, when 



| arg / 1 < TT, | arg (ty^T) | 



Consequently \f(ty }e~^dt taken round an arc of a circle of radius N terminated 

 by the points N exp { (|TT-|-X) } tends to zero as N*oo; cmc/ therefore, by 

 CAUCHY'S theorem, we may deform the path of integration and get 



/ ^)= f f(ty^ 



.'(-i^+w 



Now make y move from y^ to y u e' M , and we get 



F(y^)= f 



. . -(-J^+J 



Writing te- 3n for ^, we get 



Consequently 





Now co, ls ider )/(,,,,) ,-l-rf, take,, rouud an arc of a circle of radius N terminated 

 by the points 



N exp {(fTT+iX) i], N exp {(^._jx)}. 

 On this arc, which we call T, 



' 



Consequently on T, 



Therefore I/WK A exp {- | 



pJ snce 



<Aj p exp{-JN}d|f| J 



;ANJ7r+ 

 and this expression tends to zero as N* . 



