Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 1 / 3F / c

Solution

d_{D} (0, r e^{i θ}) = \int_{0}^{r} \frac{2 d t}{1 - t ^{2}} = lo g \frac{1 + r}{1 - r} = 2 artanh r .

One can also verify that it minimizes length of a curve: along any piecewise smooth curve from zero to radius

r

, the Euclidean speed is at least the absolute radial speed. Integrating

2∣ d r ∣/ (1 - r^{2})

is at least the absolute change in

lo g ((1 + r) / (1 - r))

, with equality on the radial segment. Thus the answer is independent of

θ

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