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**Unformatted text preview: **Figure 7.32. Orientation of Domain Boundary. Theorem 7.48. If f ( z ) is analytic on a bounded domain C , then contintegraldisplay f ( z ) dz = 0 . (7 . 118) Proof : If we apply Greens Theorem to the two real line integrals in (7.109), we find contintegraldisplay udx v dy = integraldisplay integraldisplay parenleftbigg v x u y parenrightbigg = 0 , contintegraldisplay v dx + udy = integraldisplay integraldisplay parenleftbigg u x v y parenrightbigg = 0 , both of which vanish by virtue of the CauchyRiemann equations (7.18). Q.E.D. If the domain of definition of our complex function f ( z ) is simply connected, then, by definition, the interior of any closed curve C is contained in , and hence Cauchys Theorem 7.48 implies path independence of the complex integral within . Corollary 7.49. If f ( z ) is analytic on a simply connected domain C , then its complex integral integraldisplay C f ( z ) dz for C is independent of path. In particular, contintegraldisplay C f ( z ) dz = 0 (7 . 119) for any closed curve C . Remark : Simple connectivity of the domain is an essential hypothesis our evalua- tion (7.114) of the integral of 1 /z around the unit circle provides a simple counterexample to (7.119) in the non-simply connected domain =to (7....

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