In Peskin & Schroeder's book (Chapter 7.2, p.223-226), they analyze the pole of the correlation function by discarding the exponential term.
They first calculate the accurate value of the correlation function in momentum space by deriving (putting a term on time integration to ensure that it is well-defined):
$$\int d^4 x \, e^{i p \cdot x} \langle \Omega | T \{ \phi(x) \phi(z_1) \phi(z_2) \cdots \} | \Omega \rangle \tag{p.223} \\=\ldots \text{(this combines a few lines in the book)} $$ $$\\=\sum_\lambda \frac{1}{2E_{\mathbf{p}}(\lambda)} \cdot \frac{i e^{i(p^0 - E_{\mathbf{p}}(\lambda) + i\epsilon) T_+}}{p^0 - E_{\mathbf{p}}(\lambda) + i\epsilon} \langle \Omega | \phi(0) | \lambda_0 \rangle \langle \lambda_{\mathbf{p}} | T\{ \phi(z_1) \cdots \} | \Omega \rangle .\tag{7.36} $$
This is quite straight forward, but then they discard the exponential term and write:
$$\begin{aligned} \int d^4 x \, e^{i p \cdot x} \langle \Omega | T \{ \phi(x) \phi(z_1) \cdots \} | \Omega \rangle \quad &\sim_{p^0 \to +E_{\mathbf{p}}} \\ &\frac{i}{p^2 - m^2 + i \epsilon} \cdot \sqrt{Z} \, \langle \mathbf{p} | T \{ \phi(z_1) \cdots \} | \Omega \rangle. \end{aligned}\text{(7.37)}$$
Then they go on to analyze the pole of the equation and extract the S matrix element.
And here’s where I start to get confused:
Why they can simply extract the information about $S$-matrix element by analyzing the pole of the $(n+2)$-correlation and don't have to evaluate the $(n+2)$-correlation function directly?
I have read this post and I still don't understand why we can simply extract the S-matrix element by analyzing the pole, because in my case, there's no multiplying the term by $p^2-m^2$, and it seems that the argument of that post cannot be applied to my case.
