I'm confused by the meaning of the field strength renormalization on page 237 of Peskin and Schroeder. In it, they define the physical mass $m$ of the unstable particle by $$m^2-m_0^2-\operatorname{Re} M^2(m^2)=0\tag{7.58}$$ where $m_0$ is the bare mass and $M^2(m^2)$ the 1PI self energy. They then say that the pole is displaced from the real axis as the propagator is given by $$G(p)\sim \frac{iZ}{p^2-m^2-iZ\operatorname{Im} M^2(p^2)}.\tag{p.237}$$ My confusion is that the field strength renormalization $Z$ here is not equal to the residue of the pole. Given that I thought this was the definition of $Z$, what is it even supposed to mean here?
A very similar question was already asked here, but this doesn't solve the problem; it seems like Peskin and Schroeder just arbitrarily define $Z$ by expanding around the physical mass, whereas it should be equal to the residue found by expanding around the pole.
