Improper definite integration with complex bounds I am looking to prove the functional equation for theta function. Source: https://www.youtube.com/watch?v=-GQFljOVZ7I&list=PL32446FDD4DA932C9&index=12
Time about 8.00. We are to integrate:
$$\int_{-\infty+\frac{ik}{x}}^{+\infty+\frac{ik}{x}}e^{-\pi xz^2}dz$$
The argument is to change the bounds to $-\infty$ and $+\infty$ using something called Estimation Lemma or "ML-Inequality". I have absolutely no knowledge of complex analysis and integrals with complex variables. 
I've read in the comments, that one could say that $\infty+k = \infty $ for any finite number $k$, equivallently for $-\infty$ but as there is $i$, this argument seems invalid to me.
Would anyone give me some deeper explanation of what we have actually done?

This is not an answer. 
I am just showing that the naïve substitution $u=z-ik/x$ produces the correct answer to this integral with arbitrary real limits $a$ and $b$. Symmetry between the two limits is not leading to error cancellation in this case.
$$I=\int_{a+ik/x}^{b+ik/x} e^{-\pi x z^2}\,dz=\frac{\text{erf}\left(\frac{\sqrt{\pi } (b x+\text{ik})}{\sqrt{x}}\right)-\text{erf}\left(\frac{\sqrt{\pi } (a x+\text{ik})}{\sqrt{x}}\right)}{2 \sqrt{x}}$$
Using the substitution $u=z-ik/x$ gives
$$I=\int_{a}^{b} e^{-\pi x (u+ik/x)^2}\,du=\frac{\text{erf}\left(\frac{\sqrt{\pi } (b x+\text{ik})}{\sqrt{x}}\right)-\text{erf}\left(\frac{\sqrt{\pi } (a x+\text{ik})}{\sqrt{x}}\right)}{2 \sqrt{x}}$$
where erf is the error function
I am just curious as to why in this case this naïve substitution works for all real values of the limits $a$ and $b$.
