Question

52 Complex Analysis Exercises (1) Does the function w = f(2) za have an antiderivative on C? Explain your answer. (2) Is (z dz = 0 for every closed contour I in C? How do you reconcile your conclusion with Cauchy's integral theorem? (3) Compute fc Log(x+3) dz, where is the circle with radius 2. cente at the origin and oriented once in the counterclockwise direction. (4) Let I be the contour in Figure 9.5. Compute Ir ( 21)2*+1 dz. 2 + 1 Fig. 9.5 The Contour I

QUESTION 2.

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Answer 1

JANO Given integral I = 1 zdz The value of above integral is not necessarily 0 for a closed curve. If we consider C to be unit circle, then for this curve, |z=1 zrei dz=ieide Also, zreiº replacing i by -i Also, along C, o varies from 0 to 21. Thus, I = L*e* iede =I=iſde >>I=i[ol" I= 2ni = zdz = 2ni, which is clearly non zero. Explanation: According to Cauchy's theorem, ſ f(z)dz = 0 if f(z) is analytic inside and on a simple closed curve C. Here above theorem is not applicable as f(z)= Z is not analytic inside our C. In fact f(z)= ž is nowhere analytic.