ABSTRACT
Corollary 2.11 tells us that the class of complex functions that are continuous and have an antiderivative on D is an especially nice class. If f : D → C is such a function, then not only can we conclude that
f (z) dz = 0 for any
closed contour C ⊂ D, we can also conclude that the integral of f from one point to another in D is path independent. That is, the value of the integral depends only on the initial point and the terminal point, not on the path (within D) taken between them. This is significant. After all, in general for f : D → C there are infinitely many pathswithinD that connect two points in D. The answer to the questionWhen will the integral of a complex function f from one point to another not depend on the choice of contour between them? is important for both practical and theoretical reasons. In particular, the property of path independence is convenient, since it allows one to choose a different contour than the one specified (if specified) connecting the initial point to the terminal point.
