An introduction to the calculus of finite differences and by Kenneth S. Miller.

By Kenneth S. Miller.

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If w is in the unbounded component of C \ [γ] then n(γ, w) = 0. Proof Suppose that w ∈ C \ [γ]. Let δ = d(w, [γ]) = inf{|γ(t) − w| : t ∈ [a, b]}. Since [γ] is closed, δ > 0. If z ∈ Nδ (w) and t ∈ [a, b], then |(γ(t) − w) − (γ(t) − z)| = |w − z| < |γ(t) − w|. Thus n(γ, z) = n(γ − z, 0) = n(γ − w, 0) = n(γ, w), and so nγ is continuous on C \ [γ]. Since nγ is integer-valued, it is constant on each of the connected components of C \ [γ]. Let M = sup{|γ(t)| : t ∈ [a, b]}. If r > M then −r is in the unbounded connected component of C \ [γ], and [γ + r] ∩ C0 = ∅, so that n(γ, −r) = n(γ + r, 0) = 0.

The image γ([a, b]) is called the track from γ(a) to γ(b), and is denoted by [γ]. A path γ is closed if γ(a) = γ(b); we return to our starting point. A path γ : [a, b] → C is simple if γ is an injective mapping from [a, b] into C. A simple closed path γ : [a, b] → C is a closed path whose restriction to [a, b) is injective. If γ : [a, b] → X and δ : [c, d] → X are paths, and γ(b) = δ(c), the juxtaposition γ ∨ δ is the path from [a, b + (d − c)] into X defined by γ ∨ δ(x) = γ(x) for x ∈ [a, b] and γ ∨ δ(x) = δ(x + (c − b)) for x ∈ [b, b + (d − c)].

Note that continuous branches are functions on X, and not on f (X). As a particular case, if X ⊆ C∗ and f (z) = z, then a continuous branchof Arg z on X is a continuous branch of the inclusion mapping of X into C∗ . For example, the principal value mapping z → arg z is a continuous branch of Arg z on the cut plane C0 . Similarly, the mapping z → arg α z is a continuous branch of Arg z on the cut plane Cα . 1 Suppose that f : (X, τ ) → C∗ is continuous and that θ is a continuous branch of Arg f on (X, τ ), that x0 ∈ X and that t0 ∈ Arg f (x0 ).

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