Figure 31 — A closed path of integration 



Figure 32 — Two alternative closed paths 

 of integration. 



fiP)f(2)d2+ f(Q)f(2)d3^0 



Here (P) and (Q) are inserted to specify that the paths of integration pass respectively 



through the points P and Q, But 



A B 



f{Q) f{2) dz^- f (Q) f(z) dz 

 ''b ''a 



hence 



a D 



J iP)f{z)dz^ J (Q)f(z)dz 



To obtain theorem (c), connect any two contours ABDA, EFGE, by a cross-path BF, 

 as in Figure 32. Then the path BDABFEGFB is a closed contour around which, under the 

 conditions assumed in theorem (c), ^ f(z) dz = 0. But the path BF is traversed twipe, in 

 opposite directions, and hence its net contribution to the integral vanishes. The contributions 

 made by the original contours are thus equal and opposite. But the contour EFGE was trav- 

 ersed with its interior on the right, or in the negative direction; if traversed positively, its 

 contribution to the integral is reversed in sign. Hence § f{z) dz has the same value around 

 the two original contours. 



55 



