The static and dynamic properties of the single-chain molecular magnet [Co(hfac)$_2$NITPhOMe] are investigated in the framework of the Ising model with Glauber dynamics, in order to take into account both the effect of an applied magnetic field and a finite size of the chains. For static fields of moderate intensity and short chain lengths, the approximation of a mono-exponential decay of the magnetization fluctuations is found to be valid at low temperatures; for strong fields and long chains, a multi-exponential decay should rather be assumed. The effect of an oscillating magnetic field, with intensity much smaller than that of the static one, is included in the theory in order to obtain the dynamic susceptibility $chi(omega)$. We find that, for an open chain with $N$ spins, $chi(omega)$ can be written as a weighted sum of $N$ frequency contributions, with a sum rule relating the frequency weights to the static susceptibility of the chain. Very good agreement is found between the theoretical dynamic susceptibility and the ac susceptibility measured in moderate static fields ($H_{rm dc}le 2$ kOe), where the approximation of a single dominating frequency turns out to be valid. For static fields in this range, new data for the relaxation time, $tau$ versus $H_{rm dc}$, of the magnetization of CoPhOMe at low temperature are also well reproduced by theory, provided that finite-size effects are included.
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