A First Course in Complex Analysis · Problem 1

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

Mathematics — Complex Analysis · path-length

Problem

Find the length of the following paths. Draw pictures of each path and convince yourself that the lengths you computed are sensible. (a) γ(t)=3t+i\gamma(t) = 3t + i, −1≤t≤1-1 \leq t \leq 1 (b) γ(t)=i+eiπt\gamma(t) = i + e^{i\pi t}, 0≤t≤10 \leq t \leq 1 (c) γ(t)=isin⁡(t)\gamma(t) = i\sin(t), −π≤t≤π-\pi \leq t \leq \pi (d) γ(t)=t−ie−it\gamma(t) = t - ie^{-it}, 0≤t≤2π0 \leq t \leq 2\pi The last path is a cycloid, the trace of a fixed point on a wheel as it makes one rotation.

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