Now 



jj^^'"^^^ l^j ^'^^ j ('^2 ""^l) ^^2/ 



provided d(fi/dx is continuous in x. Here 0„ and cf}. denote values at opposite ends of 

 the range for x, for any given value of y, as illustrated in Figure 31. 



The integral in y is then to be carried out between the extreme limits for y, and dy is 

 is here understood to be positive. This integral can also be written 



j {4>^-<^^)dy = <^<f>dy 



where ^denotes as usual the integral taken around the contour in the counterclockwise direc- 

 tion. For (^^dy equals the corresponding (^dy in the contour integration, whereas <^ dy = - ^ dy 

 since all dy's are negative along the left-hand side of the contour. Hence 



Similarly, 



J -~ dx dy = ^ c^ dy 

 I I — dx dy = I dx i (i/*, -lA,) dy = -^i^ dx 



the sign is negative here because it is at point number 2 that dx has opposite signs in the two 



integrations. 



Hence 



4dy = - ^4r dx 



Similarly, by integrating [22d] 



-^<pdx = ~^il/dy 



From these two equations it is obvious that the right-hand member of Equation [28a] vanishes. 

 Hence ^f{z) dz ^ Q. 



Theorems (b) and (c) are corollaries of (a). 



To deduce (b), let APB and AQB denote two paths of the kind specified in (b). Then 

 APBQA is a closed path to which (a) applies, so that 



54 



