quantum field theory - Why can we simply absorb the positive coefficient of $iepsilon$ in a propagator?

As far as I know, absorbing of the positive coefficient of $i\epsilon$ in a propagator seems to be a trivial operation without even the need of justification.

In Peskin page 286, he did this:

$$k^0\rightarrow k^0(1+i\epsilon)$$ $$(k^2-m^2)\rightarrow (k^2-m^2+i\epsilon)$$

In M. Srednicki's Quantum Field Theory, page 51,

The factor in large parentheses is equal to $E^2-\omega^2+i(E^2+\omega^2)\epsilon$, and we can absorb the positive coefficient in to $\epsilon$ to get $E^2-\omega^2+i\epsilon$.

Why and does this kind of manipulation affect the final result of calculation?
Although $\frac{1}{k^2-m^2+i\epsilon k^2}-\frac{1}{k^2-m^2+i\epsilon}$ is infinitesimal, but the integration of such terms may lead to divergences, and this is my worry.

Also the presence of $k^0$ in the coefficient of $i\epsilon$ could potentially influence the poles of an integrand and consequently influence the validity of Wick Rotation.

Answer

The size of the parameter $\epsilon$ does not matter, as long as it is infinitesimally small. Rescaling it by that function does not change this. Recall that the whole procedure is just a mathematical trick which allows us to perform a contour integral over the real axis of the complex plane. The shift is really arbitrary, as long as it is small. The precise size is should not affect any results we gain from it.