By Beck M., Marchesi G., Pixton G.

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Let γr be the counterclockwise circle with center at 0 and radius r. Find γr z 2 −2z−8 2 r = 3 and r = 5. (Hint: Since z − 2z − 8 = (z − 4)(z + 2) you can find a partial fraction 1 A B decomposition of the form z 2 −2z−8 = z−4 + z+2 . ) 25. Use the Cauchy integral formula to evaluate the integral in Exercise 24 when r = 3. (Hint: The integrand can be written in each of following ways: z2 1 1 1/(z − 4) 1/(z + 2) = = = . ) 26. Compute the following integrals, where C is the boundary of the square with corners at ±4 ± 4i: (a) C (b) C (c) C (d) C ez dz.

Then f (w) = 1 2π 2π f w + reit dt . 0 Furthermore, if f = u + iv, u(w) = 1 2π 2π u w + reit dt and v(w) = 0 1 2π 2π v w + reit dt . 0 Exercises 1. 1. 2. Evaluate 1 γ z dz where γ(t) = sin t + i cos t, 0 ≤ t ≤ 2π. 3. Integrate the following functions over the circle |z| = 2, oriented counterclockwise: (a) z + z. (b) z 2 − 2z + 3. (c) 1/z 4 . (d) xy. CHAPTER 4. INTEGRATION 45 4. Evaluate the integrals γ x dz, γ y dz, γ z dz and γ z dz along each of the following paths. Note that you can get the second two integrals very easily after you calculate the first two, by writing z and z as x ± iy.

Fix a point a ∈ G and let F (z) = f γz where γz is any smooth curve from a to z. We should make sure that F is well defined: Suppose δz is another smooth curve from a to z then γz − δz is closed and G-contractible, as G is simply connected. 5 0= f− f= γz −δz γz f δz which means we get the same integral no matter which path we take from a to z, so F is a well-defined function. It remains to show that F is a primitive of f : F (z + h) − F (z) 1 = lim h→0 h→0 h h f− F (z) = lim γz+h f . γz Now let δ be a smooth curve in G from z to z + h.

### A first course in complex analysis by Beck M., Marchesi G., Pixton G.

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