50 



Proceedings of the Royal Irish Academy. 



where n is any integer. It follows that the subject of integration in Jl''ha.s 

 values which, at corresponding points on opposite sides of the cut to this par- 

 ticular singularity, differ by 2(/'(«') ; thus the integrals Jt taken along the 

 two sides of the cut combine to ffive the value 



Wo 



2i 



or 2i|/(«-„;-/(<^o+ ^O}- 



(40) 



cL + iC 



The integral for the circumference of small radius t round u-^ is of the order 

 of magnitude e log i, (it being supposed that ir„ is not taken coincident with 

 any infinity oif'(w)], and this tends to the limit zero. 



lif'(w) had a logarithmic singularity, it would be dealt with in a similar 

 manner, ami would yield a term proportional to 

 the integral of 



log sin |irX"' (iVg - v:)} 

 along the corresponding straight cut. 

 A simple pole oif'{w), sa}' at 



10 = if, = ^i + 1./.1, 



would correspond to f'(w) taking the form 



P(«<- - ?'--•,)-' for w - u'l 



infinitesimally small, F being a constant. A 



complete circuit round u\ leaves the value of 



the function unaltered, so the integrals along 



the two sides of the cut cancel one another. 



This is equally true for infinities of higher integral order. But infinities of 



fractional order are branch-points, and for such the integrals along the two 



sides of the cut do not cancel one another, but give an integral which is not 



generally more susceptible of direct evaluation than the integral </. For this 



reason the method of contour integration is not likely to be helpful when 



/'(«') has branch-points. 



The integral along the circumference round i'\ of infinitesimal radius e, 

 taken in the conventionally negative sense, is readily seen to have the limit 



- 2iF log sin j irX" ' (wo - i'-'i) i . 

 The treatment of an infinity of higher integral order is sufiiciently 

 illustrated by considering the case in which /'("')> ^'^^ ^■' ~ '-'' small, tends to 

 the form 



Q(H--w,y' + F{iv-,r,)-'. 

 Near 2i„ approximately. 



a+X 



FiGVltK 3. 



- log sin ^ (w„ - w) = - log sin ^ ('^"o - "-'') 



TT A TT A 



II- - a\ n 



; cot - ((i-o - M-,), 



