  This paper shows the strong converse and the dispersion of memoryless
channels with cost constraints and performs refined analysis of the third order
term in the asymptotic expansion of the maximum achievable channel coding rate,
showing that it is equal to $\frac 1 2 \frac {\log n}{n}$ in most cases of
interest. The analysis is based on a non-asymptotic converse bound expressed in
terms of the distribution of a random variable termed the $\mathsf b$-tilted
information density, which plays a role similar to that of the $\mathsf
d$-tilted information in lossy source coding. We also analyze the fundamental
limits of lossy joint-source-channel coding over channels with cost
constraints.
