Complex Variables by Robert B. Ash, W.P. Novinger

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By Robert B. Ash, W.P. Novinger

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Define Ω = {z ∈ C \ γ ∗ : n(γ, z) = 0}. 5). 2), then g(w, z) = (f (w) − f (z))/(w − z) since z ∈ / γ ∗ . 3. CAUCHY’S THEOREM 11 since n(γ, z) = 0 for z ∈ Ω . The above computation shows that we can define a function h on C by    γ g(w, z) dw if z ∈ Ω h(z) =   f (w) dw if z ∈ Ω . 3), h is analytic on Ω. Thus h is an entire function. 5), hence z ∈ Ω . (w) Consequently, h(z) = γ fw−z dw → 0 as |z| → ∞. 2), h ≡ 0. ∗ Now if z ∈ Ω \ γ we have, as at the beginning of the proof, 0 = h(z) = g(w, z) dw = γ γ f (w) dw − 2πin(γ, z)f (z) w−z proving (ii).

Show that f has an analytic square root, but not an analytic logarithm, on Ω. 3. Let f be an analytic zero-free function on Ω. Show that the following are equivalent. (a) f has an analytic logarithm on Ω. (b) f has an analytic k-th root on Ω (that is, an analytic function h such that hk = f ) for every positive integer k. (c) f has an analytic k-th root on Ω for infinitely many positive integers k. 4. 4d), the “generalized dog-walking theorem”. Let γ1 , γ2 : [a, b] → C be closed curves such that |γ1 (t) − γ2 (t)| < |γ1 (t)| + |γ2 (t)| for all t ∈ [a, b].

But then z0 ∈ / L, contradicting out assumption. Thus, it must be the case that an = 0 for all n, so that f ≡ 0 on D(z0 , r). Consequently, D(z0 , r) ⊆ L, proving that L is open in Ω. 4. 8 23 The Identity Theorem Suppose f is analytic on the open connected set Ω. Then either f is identically zero on Ω or else Z(f ) has no limit point in Ω. Equivalently, if Z(f ) has a limit point in Ω, then f is identically 0 on Ω. Proof. 7), the set L of limit points of Z(f ) is both open and closed in Ω. Since Ω is connected, either L = Ω, in which case f ≡ 0 on Ω, or L = ∅, so that Z(f ) has no limit point in Ω.

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