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Complex Theorem for Continuous Anti-derivative

Although Corollary 2 does not apply to the function 1 / (z — z_0) in the plane punctured at z_0, Theorem 6 can be used as follows to show that
(see attached 1)
for any circle C traversed once in the positive direction surrounding the point z_0. Introduce a horizontal branch cut from z_0 to infinity as in Fig. 4.25. In the resulting "slit plane" the function 1 / (z — z_0) has the anti-derivative Log (z — z_0). Apply Theorem 6 to compute the integral along the portion of C from alpha to beta as indicated in Fig. 4.25. Now let alpha and beta approach the point t on the cut to evaluate the given integral over all of C.
(see attached 1)
Theorem 6. Suppose that the function f (z) is continuous in a domain D and has an anti-derivative F(z) throughout D; i.e., dF(z)Idz = f (z) for each z in D. Then for any contour gamma lying in D, with initial point z_I and terminal point z_T, we have
(see attached 2).


Solution Summary

The solution assists with using the complex theorem for continuous anti-derivative and computing the integrals.