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Integration with variables Notes_Part_19

# Integration with variables Notes_Part_19 - Figure 7.32...

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Figure 7.32. Orientation of Domain Boundary. Theorem 7.48. If f ( z ) is analytic on a bounded domain Ω C , then contintegraldisplay Ω f ( z ) dz = 0 . (7 . 118) Proof : If we apply Green’s Theorem to the two real line integrals in (7.109), we find contintegraldisplay Ω udx vdy = integraldisplay integraldisplay Ω parenleftbigg ∂v ∂x ∂u ∂y parenrightbigg = 0 , contintegraldisplay Ω vdx + udy = integraldisplay integraldisplay Ω parenleftbigg ∂u ∂x ∂v ∂y parenrightbigg = 0 , both of which vanish by virtue of the Cauchy–Riemann 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 Ω is contained in Ω, and hence Cauchy’s Theorem 7.48 implies path independence of the complex integral within Ω. Corollary 7.49. If f ( z ) is analytic on a simply connected domain Ω C , then its complex integral integraldisplay C f ( z ) dz for C Ω is independent of path. In particular, contintegraldisplay C f ( z ) dz = 0 (7 . 119) for any closed curve C Ω . 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 Ω = C \ { 0 } . Interestingly, this result also admits a converse: a continuous complex-valued function that satisfies (7.119) for

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Integration with variables Notes_Part_19 - Figure 7.32...

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