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ch11 - Chapter Eleven Argument Principle 11.1 Argument...

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Chapter Eleven Argument Principle 11.1. Argument principle. Let C be a simple closed curve, and suppose f is analytic on C . Suppose moreover that the only singularities of f inside C are poles. If f z 0 for all z C , then  f C is a closed curve which does not pass through the origin. If t , t is a complex description of C , then t f t  , t is a complex description of . Now, let’s compute C f z f z dz f t  f t  t dt . But notice that t f t  t . Hence, C f z f z dz f t  f t  t dt t t dt 1 z dz n 2 i , where | n | is the number of times ”winds around” the origin. The integer n is positive in case is traversed in the positive direction, and negative in case the traversal is in the negative direction. Next, we shall use the Residue Theorem to evaluate the integral C f z f z dz . The singularities of the integrand f z f z are the poles of f together with the zeros of f . Let’s find the residues at these points. First, let Z z 1 , z 2 , , z K be set of all zeros of f . Suppose the order of the zero z j is n j . Then f z z z j n j h z and h z j 0. Thus, 11.1

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f z f z z z j n j h z n j z z j n j 1 h z z z j n j h z h z h z n j z z j . Then
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ch11 - Chapter Eleven Argument Principle 11.1 Argument...

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