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

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Use γ(t) = a + reit . 5. Evaluate γ e3z dz for each of the following paths: (a) The straight line segment from 1 to i. (b) The circle |z| = 3. (c) The parabola y = x2 from x = 0 to x = 1. 6. Evaluate γ z 2 dz where γ is the parabola with parametric equation γ(t) = t+it2 , 0 ≤ t ≤ 1. 1 1 7. Evaluate γ z 2 dz where γ is the unit circle and z 2 is the principal branch. You can use the parameterization γ(θ) = eiθ for −π ≤ θ ≤ π, and remember that the principal branch is √ 1 defined by z 2 = reiθ/2 if z = reiθ for −π ≤ θ ≤ π.

How does this 6. Compute the following integrals; use the principal value of z i . ) (a) γ1 z i dz where γ1 (t) = eit , − π2 ≤ t ≤ π2 . z i dz where γ2 (t) = eit , (b) γ2 π 2 ≤t≤ 3π 2 . 7. Suppose f is continuous on C and lim|z|→∞ f (z) = 0. Show that f is bounded. (Hint: From the definition of limit at infinity (with = 1) there is R > 0 so that |f (z) − 0| = |f | (z) < 1 if |z| > R. ) CHAPTER 5. CONSEQUENCES OF CAUCHY’S THEOREM 56 8. Let p be a polynomial of degree n > 0. Prove that there exist complex numbers c, z1 , z2 , .

These identities have the flavor of mean values. Let’s summarize them in the following statement, which is often called a mean-value theorem. 10. Suppose f is analytic on and inside the circle z = w + reit , 0 ≤ t ≤ 2π. 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.

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

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