  We prove that the pressure (or free energy) of the finite range ferromagnetic
Ising model on $\mathbb{Z}^d$ is analytic as a function of both the inverse
temperature $\beta$ and the magnetic field $h$ whenever the model has the
exponential weak mixing property. We also prove the exponential weak mixing
property whenever $h\neq 0$. Together with known results on the regime
$h=0,\beta<\beta_c$, this implies both analyticity and weak mixing in all the
domain of $(\beta,h)$ outside of the transition line $[\beta_c,\infty)\times
\{0\}$. The proof of analyticity uses a graphical representation of the Glauber
dynamic due to Schonmann and cluster expansion. The proof of weak mixing uses
the random cluster representation.
