The application of the conventional saddle-point approximation to condensed
Bose gases is thwarted by the approach of the saddle-point to the ground-s
tate singularity of the grand canonical partition function. We develop and
test a variant of the saddle-point method which takes proper care of this c
omplication, and provides accurate, flexible, and computationally efficient
access to both canonical and microcanonical statistics. Remarkably, the er
ror committed when naively employing the conventional approximation in the
condensate regime turns out to be universal, that is, independent of the sy
stem's single-particle spectrum. The new scheme is able to cover all temper
atures, including the critical temperature interval that marks the onset of
Bose-Einstein condensation, and reveals in analytical detail how this onse
t leads to sharp features in gases with a fixed number of particles. In par
ticular, within the canonical ensemble the crossover from the high-temperat
ure asymptotics to the condensate regime occurs in an error-function-like m
anner; this error function reduces to a step function when the particle num
ber becomes large. Our saddle-point formulas for occupation numbers and the
ir fluctuations, verified by numerical calculations, clearly bring out the
special role played by the ground state. (C) 1999 Academic Press.