 # 1. Let f be defined by f(z) = Log(E), and let C be the circle [z| =...

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1. Let f be defined by f(z) = Log(E), and let C be the circle [z| = 1 taken in the positive direction. Without evaluating the integral prove that 2. Can the Cauchy-Goursat Theorem be applied to show that Scf(z)dz= = 0, if f(z) = csc(z), and C is the ellipse z2 - = 3. Can the Cauchy-Goursat Theorem be applied to show that Scf(z)dz: = 0, if f(z) = in and C is the circle [z| = 1 7 4. Evaluate Sc f(z)dz, where f(z) = :2+1 and C is the circle - 2il = 2 taken in the positive direction. Express your answer in the form . + iy where x and y are real. 5. Let f be defined on the complex plane by f(z) = z if Re(z) > 0, and f(z) = ² if Re(z) < 0. Let C denote the circle [z| = 1 taken in the positive direction. Evaluate So f(z)dz. Express your answer in the form x+iy where x and y are real.

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