I am dealing with the following integral defined in Eq. $(40)$. I am kind of sure that steps from Eqs. $(40)$ to $(41)$, $(41)$ to $(42)$ are both fine.
I tested on Mathematica step from Eq. $(42)$ to Eq. $(43)$ seems giving different values. But I could not pin down what is going to wrong.
I am still thinking of using analytic continuation to pull the integral contour for $w$ down from $[-\infty+ip, \infty+ip]$ to $[-\infty, \infty]$. In the other words, one can apply the residue theorem for the contour $-\infty \to \infty \to \infty+ip \to -\infty+ip \to -\infty$. This argument seems to justify that Eqs. $(42)$ and $(43)$ are indeed equal.
Anyone has idea on what is going there? Thanks a lot in advance!
$$ \begin{align*} I\left( p, y \right) &= \int_{-\infty}^{\infty} \exp\left[ -z^{2} - 2 \cdot i \cdot p \cdot z + \left| z - y \right| - y^{2} \right] \operatorname{d}z \tag{40}\\ &= e^{-p^{2}} \cdot \int_{-\infty}^{\infty} \exp\left[ -\left( z + i \cdot p \right)^{2} + \left| z - y \right| - y^{2} \right] \operatorname{d}z \tag{41}\\ &= e^{-p^{2}} \cdot \int_{-\infty + i \cdot p}^{\infty + i \cdot p} \exp\left[ -w^{2} + \left| w - y - i \cdot p \right| - y^{2} \right] \operatorname{d}w \tag{42}\\ &= e^{-p^{2}} \cdot \int_{-\infty}^{\infty} \exp\left[ -w^{2} + \left| w - y - i \cdot p \right| - y^{2} \right] \operatorname{d}w \tag{43}\\ \end{align*} $$