The ratio of the arc to the chord of an analytic curve need not approach unity
Author: Edward Kasner
Journal: Bull. Amer. Math. Soc. 20 (1914), 524-531
If $P$ is a fixed point on a curve and $Q$ is a point which
approaches $P$ along the curve, the limit of the ratio of the
arc $PQ$ to the chord $PQ$ is unity. While this statement is
frequently made without reservation, it is easy, as in most
analogous statements, to construct exceptions in the domain
of real functions: by making the curve sufficiently crinkly
the limit may become say two, or any assigned number
greater than unity.
The object of this note, however, is to point out the necessity
for reservation even in the domain of (complex) analytic
curves. The limit may then be less than unity. For example,
in the imaginary parabola$$y=i x+x^2$$the value of the limit in question, at the origin, is not one, but
about $0.94$ . The exact value is easily found to be $\frac23\sqrt2$.
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Posted at 2022-8-6 08:48:08
