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

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1(d): T dz 2 2 ≤ 2 · πR = z2 + 1 R R and this has limit 0 as R → ∞. On the other hand, we can parameterize the integral over S using z = t, −R ≤ t ≤ R, obtaining R dt dz = . 2 2 S z +1 −R 1 + t CHAPTER 5. CONSEQUENCES OF CAUCHY’S THEOREM 53 As R → ∞ this approaches an improper integral. Making these observations in the limit of the formula (∗∗) as R → ∞ now produces ∞ −∞ dt = π. t2 + 1 Of course this integral can be evaluated almost as easily using standard formulas from calculus. However, just a slight modification of this example leads to an improper integral which is far beyond the scope of basic calculus; see Exercise 11.

Different definitions might lead to different outcomes of ez versus exp z! 3 Named after Leonard Euler (1707–1783). html. CHAPTER 3. EXAMPLES OF FUNCTIONS 33 Exercises 1. Show that if f (z) = az+b cz+d is a M¨ obius transformation then f −1 (z) = dz−b −cz+a . 2. Show that the derivative of a M¨ obius transformation is never zero. 3. Prove that any M¨ obius transformation different from the identity map can have at most two fixed points. ) 4. 2. 5. Show that the M¨ obius transformation f (z) = onto the imaginary axis.

1. Here is the theorem on which most of what will follow is based. CHAPTER 4. 1: This square and the circle are (C \ {0})-homotopic. 4 (Cauchy’s Theorem). Suppose G ⊆ C is open, f is analytic in G, and γ1 ∼G γ2 via a homotopy with continuous second partials. Then f= γ1 f. γ2 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.

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

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