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

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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 b a f ( x ) dx . For vector fields F = ( P, Q ) in the plane we have the line integral 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 C P dx + Q dy only depends on the endpoints of C , equivalently if and only if C P dx + Q dy = 0 for every closed curve C . If P dx + Q dy = dg , and C has endpoints z 0 and z 1 , then we have the formula C P dx + Q dy = C dg = g ( z 1 ) - g ( z 0 ) . 2. If D is a plane region with oriented boundary ∂D = C , then C P dx + Q dy = 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 Green’s 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 C f ( z ) dz for any reasonable closed oriented curve C . If C is a parametrized curve given 1
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by r ( t ), a t b , then we can view r ( t ) as a complex-valued curve, and then C f ( z ) dz = 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 C f ( z ) dz = 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 C z 2 dz = 1 0 ( t + 2 t 2 i ) 2 (1 + 4 ti ) dt = 1 0 ( t 2 - 4 t 4 + 4 t 3 i )(1 + 4 ti ) dt = 1 0 [( 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 0 = - 11 / 3 + ( - 2 / 3) i. For another example, let let C be the unit circle, which can be efficiently parametrized as r ( t ) = e it = cos t + i sin t , 0 t 2 π , and let f ( z ) = ¯ z . Then r ( t ) = - sin t + i cos t = i (cos t + i sin t ) = ie it . Note that this is what we would get by the usual calculation of d dt e it . Then C ¯ z dz = 2 π 0 e it · ie it dt = 2 π 0 e - it · ie it dt = 2 π 0 i dt = 2 πi. 3.2 Cauchy’s theorem Suppose now that C is a simple closed curve which is the boundary ∂D of a region in C . We want to apply Green’s theorem to the integral C f ( z ) dz .
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