  The feasibility of extrapolation of completely monotone functions can be
quantified by examining the worst case scenario, whereby a pair of completely
monotone functions agree on a given interval to a given relative precision, but
differ as much as it is theoretically possible at a given point. We show that
extrapolation is impossible to the left of the interval, while the maximal
discrepancy to the right exhibits a power law typical for extrapolation of
similar classes of complex analytic functions. The power law exponent is
derived explicitly, and shows a precipitous drop immediately beyond the right
end-point, with a subsequent decay to zero inversely proportional to the
distance from the interval. The local extrapolation problem, where the worst
discrepancy from a given completely monotone function is sought, is also
analyzed. In this case explicit and easily verifiable optimality conditions are
derived, enabling us to solve the problem exactly for a single decaying
exponential. In the general case, our approach leads to a natural algorithm for
computing solutions to the local extrapolation problem numerically. The methods
developed in this paper can easily be adapted to other classes of analytic
functions represented as integral transforms of positive measures with analytic
kernels.
