Expression in John R. Taylor Scattering theory book

In the book Scattering theory by John R. Taylor He has the following expression at page 138

$$
i lim _{epsilon downarrow 0} int_{0}^{infty} d tleftlanglemathbf{p}^{prime}right|V e^{ileft(E_{p}^{prime}+E_{p}+i epsilon-2 Hright) t}|mathbf{p}rangle=1 / 2 lim _{epsilon downarrow 0}leftlanglemathbf{p}^{prime}right|V Gleft(frac{E_{p^{prime}}+E_{p}}{2}+i epsilonright)|mathbf{p}rangle tag 1
$$

where $G$ is the green operator defined by
$G(z)=(z-H)^{-1}$

But for me we should have
$$
i lim _{epsilon downarrow 0} int_{0}^{infty} d tleftlanglemathbf{p}^{prime}right|V e^{ileft(E_{p}^{prime}+E_{p}+i epsilon-2 Hright) t}|mathbf{p}rangle=1 / 2 lim _{epsilon downarrow 0} leftlanglemathbf{p}^{prime}right|V Gleft(frac{E_{p^{prime}}+E_{p}+i epsilon}{2}right)|mathbf{p}rangle tag 2
$$
Since we have that

$$int dt e^{ileft(E_{p}^{prime}+E_{p}+i epsilon-2 Hright) t} =int dt e^{2ileft(frac{E_{p^{prime}}+E_{p}+i epsilon}{2}+ Hright) t} =frac{1}{2ileft(frac{E_{p^{prime}}+E_{p}+i epsilon}{2}+ Hright)}e^{2ileft(frac{E_{p^{prime}}+E_{p}+i epsilon}{2}+ Hright) t} tag 3$$
And from $(3)$ we have expression $(2)$.
I am not seeing how did he obtain expression $1$