  We present improved numerical approximations to the exact Poissonian
confidence limits for small numbers n of observed events following the approach
of Gehrels (1986). Analytic descriptions of all parameters used in the
approximations are provided to allow their straightforward inclusion in
computer algorithms for processing of large data sets. Our estimates of the
upper (lower) Poisson confidence limits are accurate to better than 1% for
n<100 and values of S, the derived significance in units of Gaussian standard
deviations, of up to 7 (5). In view of the slow convergence of the commonly
used Gaussian approximations toward the correct Poissonian values, in
particular for higher values of S, we argue that, for n<40, Poissonian
statistics should be used in most applications, unless errors of the order of,
or exceeding, 10% are acceptable.
