,,,,; MR. O. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. 



x\*br+* where X> 0; and let the region in which f(x) is 



-- x ...i.-~~ distance from the 



y c w <* P* <" 



boundary of the sector does not exceed 2A, where A > 0. 

 Le, x . fc Hri >/"", I*"*** Ae ctor and * 



to 



the sector, then- exists an inequality of the form 



| <Aexp{-HK 



where A is a constant independent of x. 

 Then the function f(x) is identically zero 

 Let x. be any point within or on the boundary of the region \VgX\* 



Then 



round an appropriate contour. 



Since any singularity of f(x) is, at a distance, greater than or equal to 2A from x , 

 we may take the contour to be a circle of radius A with x* as centre. 



We then get, without difficulty, 



j/w(ab)|SlA-.M 



where M is the greatest value of |/()| on the contour, and /<"> denotes the n th 

 differential coefficient of /. 

 But on the contour 



|/(0|<Aexp{-|m 



<Aexp{A-|.ro|} since j<|a|ob|-A. 



That is to say, if | arg x 1 2 ir + X, 



Let us denote the integral of a function taken along a line from the origin to 

 infinity, inclined at an angle to the real axis, by the symbol ( . 

 Consider the function F defined by the equation 



The function F (y) is analytict in the interior of the region given by | arg y \ < TT + X 

 (but possibly it is not analytic on the boundary of the region). 



* The proof of the lemma is suggested by a paper by PHRAGMEN, ' Acta Mathematics,' vol. 28, 

 pp. 351-368. His paper, however, deals with integral functions, whereas we know nothing at all about 

 the behaviour of f(x) outside a certain sector of the plane. 



t The condition given by BROMWICH, ' Theory of Infinite Series,' p. 438, is satisfied by defining his 

 function M (/) by the equation M(/) = 



