By Beck M., Marchesi G., Pixton G.
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Extra resources for A first course in complex analysis
2 Remarks. 1. The condition on the smoothness of the homotopy can be omitted, however, then the proof becomes too advanced for the scope of these notes. In all the examples and exercises that we’ll have to deal with here, the homotopies will be ‘nice enough’ to satisfy the condition of this theorem. 2. It is assumed that Johann Carl Friedrich Gauß (1777–1855)1 knew a version of this theorem in 1811 but only published it in 1831. Cauchy published his version in 1825, Weierstraß2 his in 1842. Cauchy’s theorem is often called the Cauchy–Goursat Theorem, since Cauchy assumed that the derivative of f was continuous, a condition which was first removed by Goursat3 .
1) a For a function which takes complex numbers as arguments, we integrate over a curve γ (instead of a real interval). Suppose this curve is parametrized by γ(t), a ≤ t ≤ b. 1) should come as no surprise. 1. Suppose γ is a smooth curve parametrized by γ(t), a ≤ t ≤ b, and f is a complex function which is continuous on γ. Then we define the integral of f on γ as b f= γ f (z) dz = γ f (γ(t))γ (t) dt . a This definition can be naturally extended to piecewise smooth curves, that is, those curves γ whose parametrization γ(t), a ≤ t ≤ b, is only piecewise differentiable, say γ(t) is differentiable on the intervals [a, c1 ], [c1 , c2 ], .
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. (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.