terms of two real line integrals in which the variables of integration are x and y. Thus, if 

 f{z) = cf){x,y) + i xjj {x,y) where ^ and xp are real functions, since dz= dx + idy, 



I f(z) dz = I ((f>dx - ip dy) + i \ {ilr dx + (fj dy) [28a] 



Here dx and dy may be interpreted as components of successive elements dz, and values of 

 and il/ are to be taken at points lying on the corresponding elementary segments of the path. 

 Negative values of dx and dy may occur as well as positive values. 



The integral of f{z) along a closed curve is often denoted by ^ f(z) dz. This symbol 

 will be understood to imply that the curve is traversed in the positive or counterclockwise 

 direction, that is, in such a direction that its interior lies on the left. 



29. THE CAUCHY INTEGRAL THEOREM 



As in the case of integrals with respect to real variables, reversing the direction of 

 integration along the path reverses the sign of the value of f f{z) dz. But integrals of f{z) 

 along different end points, such as APB and AQB in Figure 30, may or may not be equal; and 

 ^/(s) dz taken around a closed path or contour, such as hi in Figure 30, may or may not vanish. 

 If an integral around a contour does not vanish, its sign is changed if the direction of integra- 

 tion around the contour is reversed. 



The following important theorems can, however, be proved. The first two taken together 

 are known as Cauchy's integral theorem. 



(a) If f{z) is regular at all points both inside of and on a closed contour, then around the 

 contour ^/(s) dz = 0. 



(b) If f{z) is regular at all points between and on two paths joining two end points P and P\ 

 then /^ /(3) dz has the same value along both paths. 



(c) If f{z) is regular at all points between and on two closed contours of which one 

 encloses the other, then ^f(z) dz has the same value around both contours. 



In all three cases, it is also sufficient if f{z), instead of being actually regular on the 

 contour or path itself, is merely continuous from the contour or path into the region in which it 

 is required to be regular. 



Cauchy's second proof of (a) is instructive enough to be repeated here. It is open to a 

 certain logical objection, however; a more satisfactory proof can be found in books on functions 

 of a complex variable (for example, E.T. Copson ^^). 



Let the first of the Cauchy-Riemann equations or [22c] be integrated with respect to x 

 and y over the area on the 2-plane enclosed within the contour, giving 



//f-^-//^"^' 



53 



