Please answer question4 All the steps and claculations must

Please answer question4. All the steps and claculations must be shown with formula. An incomplete answer will result in a thumb down and bad rating for you. A good answer will result in a thumb up and I would really apprecaite it. Thanks

pate Exercise 2. 4. Let r, denote polar coordinates centered at . Evaluate (1/2) u(R, )di for u, c, r = R as follows: (a) u(z) = In , = (1,0), R = 1/2; (b) u(z) = x + y,L = (1,-1), R = 5. Hint: No need to change u(2) to polar coordinates nor to integrate. 5. Evaluate J u(x, y) dx dy for u and nci m

Solution

4)

a) ln|z| is holomorphic in C\\{0} the curve enclosed the region where ln|z| is holomorphic . Hence the required integral is zero by cauchy theorem.

b) if (x,y) is on the circle (2-x,-2-y) is also on the circle the diametrically opposite point. by symmetry the integration will be zero

(x+y) + (2-x + -2-y) = 0

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