In the book Scattering theory by John R. Taylor He has the following expression at page 138
$$ i \lim _{\epsilon \downarrow 0} \int_{0}^{\infty} d t\left\langle\mathbf{p}^{\prime}\right|V e^{i\left(E_{p}^{\prime}+E_{p}+i \epsilon-2 H\right) t}|\mathbf{p}\rangle=1 / 2 \lim _{\epsilon \downarrow 0}\left\langle\mathbf{p}^{\prime}\right|V G\left(\frac{E_{p^{\prime}}+E_{p}}{2}+i \epsilon\right)|\mathbf{p}\rangle \tag 1 $$
where $G$ is the green operator defined by $G(z)=(z-H)^{-1}$
But for me we should have $$ i \lim _{\epsilon \downarrow 0} \int_{0}^{\infty} d t\left\langle\mathbf{p}^{\prime}\right|V e^{i\left(E_{p}^{\prime}+E_{p}+i \epsilon-2 H\right) t}|\mathbf{p}\rangle=1 / 2 \lim _{\epsilon \downarrow 0} \left\langle\mathbf{p}^{\prime}\right|V G\left(\frac{E_{p^{\prime}}+E_{p}+i \epsilon}{2}\right)|\mathbf{p}\rangle \tag 2 $$ Since we have that
$$\int dt e^{i\left(E_{p}^{\prime}+E_{p}+i \epsilon-2 H\right) t} =\int dt e^{2i\left(\frac{E_{p^{\prime}}+E_{p}+i \epsilon}{2}+ H\right) t} =\frac{1}{2i\left(\frac{E_{p^{\prime}}+E_{p}+i \epsilon}{2}+ H\right)}e^{2i\left(\frac{E_{p^{\prime}}+E_{p}+i \epsilon}{2}+ H\right) t} \tag 3$$ And from $(3)$ we have expression $(2)$. I am not seeing how did he obtain expression $1$