  In this paper we provide new compact integral expressions and associated
simple asymptotic approximations for converse and achievability bounds in the
finite blocklength regime. The chosen converse and random coding union bounds
were taken from the recent work of Polyanskyi-Poor-Verdu, and are investigated
under parallel AWGN channels, the AWGN channels, the BI-AWGN channel, and the
BSC. The technique we use, which is a generalization of some recent results
available from the literature, is to map the probabilities of interest into a
Laplace integral, and then solve (or approximate) the integral by use of a
steepest descent technique. The proposed results are particularly useful for
short packet lengths, where the normal approximation may provide unreliable
results.
