These theorems are of enormous value in the evaluation of complex integrals. The 

 evaluation can often be rendered very easy by deforming the path into a suitable shape; the 

 path can be deformed at will so long as it is not deformed past any point at which f{z) ceases 

 to be regular. 



30. SINGULAR POINTS AND RESIDUES 



An important case in applications of the theory is that in which f{z) is regular through- 

 out a certain region S except at one or more internal points. These excluded points may be 

 singular points, or they may be points at which nothing is known or assumed about the function. 



Suppose that S contains one excluded point. Then ^ f{z) dz has the same value for 



all closed curves lying in S which encircle this point once. This follows from theorem (c) in 



Section 29, in view of the fact that no excluded point occurs either between or on the two 



curves. The number 



1 

 - — : ^f{z)dz 



'2ni 



is called the residue of the function f{z) at the excluded point. 



If more than one excluded point occurs in S, the value of ^f(z)dz around a curve en- 

 circling any finite number of them is 27ri times the sum of the residues of (2) at the encircled 

 points. This is proved by deforming the original curve until it consists of separate curves 

 encircling one singular point each and connected by paths that are traversed twice, as illus- 

 trated in Figure 33, where Q and R represent two excluded points and the outer curve is the 

 original one. The connecting paths contribute nothing to ^/(s) dz taken around the combined 

 curve. 



As an example, consider 



/(3)= 



(z-ar 

 where 71 is a positive integer and a and k are constants. This function has one singularity, at 

 B=a. Let the path of integration be a circle of radius fi about z=a as center. Then, for values 

 of 2 on the circle, js-fll = R, and, if 6 is the amplitude of z-a, 



z-a = Re'^, dz - iRe*^ dd 

 since R is constant along the circle. Thus 



r kdz ^^C JRe^^ dQ ^,:^ffi-n r e'(i-">' 

 J (z-aY J (Re'^)" Jo 



If n>l, 



/ 



-^'^^ -ikR^-" / =0 



(^-^)" Hl-n) le 



56 



