  Analytic continuation is ill-posed, but becomes merely ill-conditioned
(although with an infinite condition number) if it is known that the function
in question is bounded in a given region of the complex plane. In an annulus,
the Hadamard three-circles theorem implies that the ill-conditioning is not too
severe, and we show how this explains the effectiveness of Chebfun and related
numerical methods in evaluating analytic functions off the interval of
definition. By contrast, we show that analytic continuation is far more
ill-conditioned in a strip or a channel, with exponential loss of digits of
accuracy at the rate $\exp(-\pi x/2)$ as one moves along. The classical
Weierstrass chain-of-disks method loses digits at the faster rate $\exp(-e\kern
.3pt x)$.
