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Unformatted text preview: 3 Contour integrals and Cauchy’s 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 1forms 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 complexvalued. Then all of the above still makes sense, and in particular Green’s theorem is still true. We will be interested in the following integrals. Let dz = dx + idy , a complex 1form, 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 complexvalued 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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This note was uploaded on 10/18/2011 for the course MATH S1202Q taught by Professor Peters during the Spring '11 term at Columbia.
 Spring '11
 Peters
 Calculus, Integrals

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