REGENERATION THEORY 



135 



We shall next evaluate the integral for a very large value of /. It 

 will suffice to take the C integral since the / integral approaches zero. 

 Assume originally that 1 — iv does not have a root on the imaginary 

 axis and that F{z) has the special value w'{z). The integral may be 

 written 



2wX 



Iw'ia - iu)2e''dz. 



Changing variables it becomes 



2^.//'/(' 



ii!)2e''dw, 



(36) 



(37) 



where c is a function of iv and D is the curve in the w plane which 

 corresponds to the curve C in the z plane. More specifically the 

 imaginary axis becomes the locus x = and the semicircle becomes a 

 small curve which spirals around the origin. See Fig. 2. The function 



x=o 



W -PLANE 



Fig. 2 — Representative paths of integration in the ic-plane corresponding to paths 



in Fig. 1. 



2 and, therefore, the integrand is, in general, multivalued and the 

 curve of integration must be considered as carried out over the 

 appropriate Riemann surface.'^ 



Now let the path of integration shrink, taking care that it does not 

 shrink across the pole at if = 1 and initially that it does not shrink 

 across such branch points as interfere with its passage, if any. This 

 shrinking does not alter the integral ^ because the integrand is analytic 

 at all other points. At branch points which interfere with the passage 

 of the path the branches stopped may be severed, transposed and 

 connected in such a way that the shrinking may be continued past the 

 branch point. This can be done without altering the value of the 

 integral. Thus the curve can be shrunk until it becomes one or more 

 very small circles surrounding the pole. The value of the total integral 



' Osgood, loc. cit., Kap. 8. 



* Osgood, loc. cit., Kap. 7, § 3, Satz 1. 



