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Let f z n C fz Define (be a continuous function on ) and suppose that for some fixed n n 0 and c -{0}: c as z r 0: f r ei t and F r 0, 2 r t t Then...

The question is attached, and is from Rudin's "Principles of Mathematical Analysis". Essentially, the question is this:

Let f be a continuous complex function defined in the complex plane.
Suppose there is an integer n and a nonzero complex number such that lim z^(-n)f(z)=c

Prove that f(z)=0 for at least one complex number z
(the suggested three steps of the proof are included in the attached document)
It is suggested that we use the index/winding number of a curve to prove this.
Let f ˛ C H C L (be a continuous function on C ) and suppose that for some fixed n ˛ N H n > 0 L and c ˛ C -{0}: z - n f H z L c as z fi ¥ Define " r > 0 : " t ˛ @ 0, 2 Π D Γ r H t L = f I r e i t M and F H r L = Ind H Γ r L = Then we observe that " r > 0 : Γ r H 2 Π L = f I r e i 2 Π M = f H r L = f I r e i 0 M = Γ r H 0 L , so Γ r is a closed contour and is continuous by the continuity of f . Assume that, " z ˛ C : f H z L 0 (The claim to be proven is that f H z L = 0 for some z ˛ C ) Then it follows that $ Μ = inf 8 f H z L : z ˛ C < > 0 and $ Μ ' = inf 8 Γ r H t L : t ˛ @ 0, 2 Π D< > 0 Then it follows that $ Δ > 0 : Γ r H t L > Δ By 8.26, we see that the index of Γ r is well-defined for each r > 0 by Ind H P r L where P r - Γ r < Δ ± 4 is a polynomial which approximates Γ r (which exists by Theorem 8.15) (in Rudin, we are told as a hint to prove the following:) Part A) Claim: Ind H Γ 0 L = F H 0 L = 0 Part B) Claim: $ R > 0 : r R ± Ind H Γ r L = n Part C) Claim: Ind H Γ r L = F H r L is a continuous function of r on @ 0, ¥ L
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Lemma .1 Let γ be the winding number and assume also that γ (t) ∈ (−∞, 0] for all t ∈ [a, b]. Then Ind(γ ) = 0.
Proof. Consider the function f : R≥0 → R, defined by f (c) = Ind(γ...

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