a ,0 



MR. 0. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. 



X,,, draw the circle |*|-! and draw the tangents to this c,rcle at 

 - y exp { + ,- (X-2*)} in the dictions of the rays arg z = {fcr+X-2*} respective], . 

 "if - 1* any point to th, riht of the curve formed by these tangents and the arc of 

 th,. Circle job^ th,ir extremities (see figure), we can find a real quantity such 

 that ]/! < X-0, and such that 



R{zexp(-tV)} > yi. 



For such a value of z we have " summed" F(z) by the equation 

 which we have just proved, viz. : 



F 



(z)= f 



.!<> 



/n o</ier wwifo, we have shown that F (z) is summable by means 

 of an integral of tlie same nature as BOEEL'S integral* 



Returning to the equations at the beginning of the section, 



z = xexp {-( + )}> /(*) = F ( 2 )- 



we see that f(x) is summable by an integral of the same nature as 

 BOREL'S integral ; the formal result is hardly worth writing down, 

 since it usually happens that a = ft, so that z = x. 

 10. We may also show, by the methods of Section 9, that if we are given a 

 function <f>(t) defined by the series 



where |a.| < A .n! />", and if <f> (t) have no singularities in the region | arg t \ S X, and it 

 when |argt|sX, |^(<)| < K exp {y|| }, where K is a constant, then the function 

 F(i) defined by the equation 



F(z)= f" 



has an asymptotic expansion in powers of 1/2 valid when |argz| < ^ir + \ 6 (where 

 8 > 0), provided that 1 2 1 be sufficiently large ; and that unity is a grade and outer 



of the expansion, and p is a radius. 

 For, when |argt|SX 6, we have 



the contour being a circle ot radius p," 1 sin 0, where p^> p; assuming that when 

 \t | a p t ~ l , !<(*) | < K, we find without difficulty that 



< n! K[exp {y\t\ +y Pl ~ l sin 0}] { Pl cosec B}". 



* From the results proved concerning <""(/) it follows that F(s) is " absolutely sunnnable." 



