A First Course in Complex Analysis · Problem 24

Consequences of Cauchy's Theorem — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · homotopy

Problem

Suppose that γ\gamma is a closed path in a region GG, parametrized by γ(t)\gamma(t), t∈[0,1]t \in [0,1], and τ\tau is a continuous increasing function from [0,1][0,1] onto [0,1][0,1]. Show that γ\gamma is GG-homotopic to the reparametrized path γ∘τ\gamma \circ \tau. Hint. Make use of τs(t)=sτ(t)+(1−s)t\tau_s(t) = s\tau(t) + (1-s)t for 0≤s≤10 \leq s \leq 1.

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