Strange use of complex analysis in Weinberg QFT 1? In the beginning of chapter 3 on scattering theory in Weinberg's QFT book there is a use of the Cauchy residual theorem that I just cannot get.
First some notation, we are looking at states that are effectively non-interacting, and are considered to be a direct product of one-particle states described by their momenta $p_i$ and a bunch of (possibly discrete) indexes $n_i$. To simplify notation the sum over all indexes and integral over all momenta is written:
$$\int ~\mathrm d\alpha\, \ldots \  \equiv \sum_{n_1\sigma_1n_2\sigma_2\cdots} \int ~\mathrm d^3p_1~ \mathrm d^3p_2\, \ldots\tag{3.1.4} $$
The energy of such a state $\alpha$ is denoted $E_\alpha$ and is the sum of 1 particle energies corresponding to the momenta: 
$$E_\alpha = p_1^0 +p_2^0+\ldots\tag{3.1.7}$$ 
Now in the book we are looking at some integrals that look like this:
$$\int ~\mathrm d\alpha~ \frac{e^{-i E_\alpha t} g(\alpha) T_{\beta \alpha}}{E_\alpha - E_\beta \pm i \epsilon}\tag{3.1.21b} $$
$g(\alpha)$ is a smooth function that is non-zero on a finite range $\Delta E$ of energies. $T_{\beta \alpha}$ can probably also be assumed to be smooth.
Now the author extends the integral to a semi-circle in the upper half plane of the energies, uses the Cauchy residual theorem and takes $t \to - \infty$ to get the result 0. My problems are:


*

*The integral is not actually over the energies, but over the momenta. The energy is a function of the momenta, so I'm sure we can do some kind of a substitution to get an integral over energy, but this integral will not be over $\mathbb{R}$ since the energy of each particle is positive. So we cannot close a semi-circle.

*To use the Cauchy theorem $g(\alpha)T_{\beta \alpha}$ must be analytic after doing all integrals except for the energy integral, but if at the same time $g(\alpha)$ is supposed to be zero outside of some finite range of energies, this is not possible.

 A: Heuristically, one approach to justifying Weinberg's application of Cauchy's formula is to treat the non-analytic integrand as the boundary behavior of a meromorphic function (kind of like Fourier series):


*

*perform all internal integrations until you are left with an integral over energy,

*solve the Cauchy-Riemann equations within the upper half plane (possibly with a singular set removed), with the integrand as a boundary condition on the real line,

*approximate the original integral using the solution just obtained (e.g. on a nearby contour within the domain of analyticity),

*estimate the approximation using the residue formula.


Of course it would be a miracle if all of these steps could be carried out without errors. Fortunately, it's easy to make sure that these errors decay in the large $t$ limit: in the integral 
$$
I=\int ~\mathrm dE~\frac{e^{-iEt}f(E)}{E-E_\beta\pm i\epsilon},
$$
the worst behavior is near $E=E_\beta$. We can eliminate this simply by separating $I$ into two parts: 
$$
I^\prime_\pm=f(E_\beta)\int~\mathrm dE~\frac{e^{-iEt}}{E-E_\beta\pm i\epsilon},\quad I_\pm^\textrm{reg} \equiv I-I^\prime_\pm.
$$
Cauchy's integral formula can be applied to show $I_\pm '$ vanishes when $t\rightarrow \mp\infty$, while $I^\textrm{reg}_\pm$ decays to zero as $|t|\rightarrow\infty$ by the usual arguments of the Riemann-Lebesgue lemma when $f(E)$ is sufficiently 'nice'. Since $|I|\leq |I^\textrm{reg}|+|I^\prime|$, it follows that $I_\pm$ vanishes (with the appropriate choice of $\pm\epsilon$.)
