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**Unformatted text preview: **Proof : Note that the integrand f ( z ) = 1 / ( z a ) is analytic everywhere except at z = a , where it has a simple pole. If a is outside C , then Cauchys Theorem 7.48 applies, and the integral is zero. On the other hand, if a is inside C , then Proposition 7.50 implies that the integral is equal to the integral around a circle centered at z = a . The latter integral can be computed directly by using the parametrization z ( t ) = a + r e i t for t 2 , as in (7.114). Q.E.D. Example 7.53. Let D C be a closed and connected domain. Let a, b D be two points in D . Then contintegraldisplay C parenleftbigg 1 z a 1 z b parenrightbigg dz = contintegraldisplay C dz z a contintegraldisplay C dz z b = 0 for any closed curve C = C \ D lying outside the domain D . This is because, by connectivity of D , either C contains both points in its interior, in which case both integrals equal 2 i , or C contains neither point, in which case both integrals are 0. The conclusion is that, while the individual logarithms are multiply-valued, their difference F ( z ) = log( z a ) log( z b ) = log z a z b (7 . 124) is a consistent, single-valued complex function on all of = C \ D . The difference (7.124) has, in fact, an infinite number of possible values, differing by integer multiples of 2 i ; the ambiguity can be resolved by choosing one of its values at a single point in . These conclusions rest on the fact that D is connected, and are not valid, say, for the twice- punctured plane C \ { a, b } . Lift and Circulation In fluid mechanical applications, the complex integral can be assigned an important physical interpretation. As above, we consider the steady state flow of an incompressible, irrotational fluid. Let f ( z ) = u ( x, y ) i v ( x, y ) denote the complex velocity corresponding...

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