An Intro to the Study of the Elements of the Diff and Int by A. Harnack

By A. Harnack

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1 Winding numbers The complex analysis that we have so far developed is essentially a straightforward development of ideas from real analysis. In the next chapter, we consider path integrals, and things will change dramatically. For this, we need to establish some of the topological properties of the complex plane C. Since the mapping (x, y) → x + iy is an isometry of R2 onto C, these properties correspond to topological properties of R2 . Suppose that (X, τ ) is a topological space and that f is a continuous mapping from X into C∗ .

6 We can rotate, dilate and translate C without changing winding numbers. 7 Suppose that γ : [a, b] → C is a path, and that w ∈ [γ]. (i) If θ ∈ R then n(eiθ γ, eiθ w) = n(γ, w). (ii) If λ > 0 then n(λγ, λw) = n(γ, w). (iii) If b ∈ C then n(γ + b, w + b) = n(γ, w). Proof More easy exercises for the reader. 8 Suppose that γ : [a, b] → C is a closed path. (i) If w ∈ [γ], and if there exists α ∈ (−π, π] such that −α ∈ Arg (γ(t) − w) for a ≤ t ≤ b, then n(γ, w) = 0. (ii) Suppose that δ : [a, b] → C is a closed path for which |δ(t) − γ(t)| < |γ(t) − w| + |δ(t) − w| for all t ∈ [a, b].

3), there exists a continuous mapping f : MR (w) → [a, b] which extends γ −1 . Thus if r = γ ◦ f , r is a retract of MR (w) onto [γ]. Let q(z) = r(z) for z ∈ U and let q(z) = z for z ∈ MR (w) \ U . ) Then q is continuous on each of the closed sets U and MR (w) \ U , and their union is MR (w), and so q is a continuous mapping of MR (w) onto MR (w) \ U . 8. 3 If γ is a simple path in C then C \ [γ] is connected. 662 The topology of the complex plane Proof Suppose, if possible, that U is a bounded connected component of C \ [γ].

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