Showing that a integral of complex variables goes to zero. I want to prove that the following $$\lim_{L \to \infty} - \int_{-L}^{L-i \omega} \frac{1}{\sqrt{2\pi}} \mathrm{e}^{-\frac{z^2}{2} } \mathrm{d} z $$ is zero. I am quite new to complex analysis so the first thing I tried was the estimation lemma but I only obtained the trivial bound. Then I parametrized the boundary as $g(t) = (1-t)(-L) +t(L-iw)$ where $t \in [0,1]$ So I obtain the integral $$\int_0^1 (2L -iw) e^{-(-L+2tL-iwt)^2} dt$$ Now I would like to conclude by saying that I have something of the form $e^{-L^2}$ times an integral of a continous function on a bounded domain. Thus sending L to infinity will give me zero. But I still have the complex values $iw$ lying around that preoccupy me. Is it possible to conclude this way? If no how could I proceed?

SInce $$\int \frac{1}{\sqrt{2\pi}} \mathrm{e}^{-\frac{z^2}{2} } \, dz=\frac{1}{2} \text{erf}\left(\frac{z}{\sqrt{2}}\right)$$ $$I=\int_{-L}^{L-i \omega} \frac{1}{\sqrt{2\pi}} \mathrm{e}^{-\frac{z^2}{2} } \,d z=\frac{1}{2} \left(\text{erf}\left(\frac{L}{\sqrt{2}}\right)+\text{erf}\left(\frac{L-i \omega }{\sqrt{2}}\right)\right)$$ Using the asymptotics of the error function, we have $$I=1-\frac{e^{-\frac{L^2}{2}} \left(1+e^{\frac{1}{2} \omega (\omega +2 i L)}\right)}{\sqrt{2 \pi } L}+O\left(\frac 1 {L^2}\right)$$