complex3 - 3 Contour integrals and Cauchys Theorem 3.1 Line...

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Unformatted text preview: 3 Contour integrals and Cauchys Theorem 3.1 Line integrals of complex functions Our goal here will be to discuss integration of complex functions f ( z ) = u + iv , with particular regard to analytic functions. Of course, one way to think of integration is as antidifferentiation. But we also have the definite integral. For a function f ( x ) of a real variable x , we have the integral Z b a f ( x ) dx . For vector fields F = ( P, Q ) in the plane we have the line integral Z C P dx + Q dy , where C is an oriented curve. Let us begin by recalling the basics of line integrals in the plane: 1. The vector field F = ( P, Q ) is a gradient vector field g , which we can write in terms of 1-forms as P dx + Q dy = dg , if and only if R C P dx + Q dy only depends on the endpoints of C , equivalently if and only if R C P dx + Q dy = 0 for every closed curve C . If P dx + Q dy = dg , and C has endpoints z and z 1 , then we have the formula Z C P dx + Q dy = Z C dg = g ( z 1 )- g ( z ) . 2. If D is a plane region with oriented boundary D = C , then Z C P dx + Q dy = ZZ D Q x- P y dxdy. 3. If D is a simply connected plane region, then F = ( P, Q ) is a gradient vector field g if and only if F satisfies the mixed partials condition Q x = P y . (Recall that a region D is simply connected if every simple closed curve in D is the boundary of a region contained in D . Thus a disk { z C : | z | < 1 } is simply connected, whereas a ring such as { z C : 1 < | z | < 2 } is not.) Suppose that P and Q are complex-valued. Then all of the above still makes sense, and in particular Greens theorem is still true. We will be interested in the following integrals. Let dz = dx + idy , a complex 1-form, and let f ( z ) = u + iv . Then we can define Z C f ( z ) dz for any reasonable closed oriented curve C . If C is a parametrized curve given 1 by r ( t ), a t b , then we can view r ( t ) as a complex-valued curve, and then Z C f ( z ) dz = Z b a f ( r ( t )) r ( t ) dt, where the indicated multiplication is multiplication of complex numbers (and not the dot product). Another notation which is frequently used is the following. We denote a parametrized curve in the complex plane by z ( t ), a t b , and its derivative by z ( t ). Then Z C f ( z ) dz = Z b a f ( z ( t )) z ( t ) dt. For example, let C be the curve parametrized by r ( t ) = t + 2 t 2 i , 0 t 1, and let f ( z ) = z 2 . Then Z C z 2 dz = Z 1 ( t + 2 t 2 i ) 2 (1 + 4 ti ) dt = Z 1 ( t 2- 4 t 4 + 4 t 3 i )(1 + 4 ti ) dt = Z 1 [( t 2- 4 t 4- 16 t 4 ) + i (4 t 3 + 4 t 3- 16 t 5 )] dt = t 3 / 3- 4 t 5 + i (2 t 4- 8 t 6 / 3)] 1 =- 11 / 3 + (- 2 / 3) i....
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complex3 - 3 Contour integrals and Cauchys Theorem 3.1 Line...

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