482 MR. IT. A. WEBB ON THE CONVERGENCE OF 



Consider the particular integral of (1) defined, when z = a, by 



where a and ft are constants, and both series converge if | k \ > 11. 

 The expression 



*-(+> + &+**+... ad infi) 



v A A / 



(2), 



where &>, (ft, x/> are functions of z independent of k, which can be constructed as a formal 

 representation of this integral, is for values of /; such that \k\ >R, and for values of 

 z within the region S, a convergent series and consequently a true representation of 

 the integral considered. 



Further, when k is very large, 



is an approximate value of the integral, whether the integral and the approximate 

 value increase indefinitely with k or not. 



This proposition has been proved by HORN* for functions of a real variable 2 ; the 

 proof for functions of a complex variable is similar ; a brief outline is as follows : 



Let -, = e u "< , r 2 = ?"""/ be two independent integrals of the equation 



Write 



u u' u" 



v\ v'\ 



Then 

 where 



E() = n'+Tu, 



S and T being functions of z and k, developable in convergent series of powers of k \ 



if |*| >R 



* ' Mjitlieiuntische Aniialcn,' vol. 52, p. 345 (1899). 



