Fourier transform of a gaussian with Cauchy theorem

Question

I'am trying to find the Fourier transform of a Gaussian function using the Cauche theorem. But without success, I found a course online but the demonstration seems false.

The demonstration is here

I absolutely don't understand why :

$$ \int^0_{-b} \mid e^{-(-a+yj)^2} \mid dy+ \int^0_{-b} \mid e^{-(a+yj)^2} \mid dy = 2 e^{-a^2}\int^{0}_{-b}e^{y^2}dy$$

And why $$ \leq 2 e^{-a^2}e^{-b^2} $$

I'am pretty sure, this is a mistake from the author but just want a confirmation.

And the next question is, how can I demonstrate that : $$\int^0_{-b} \mid e^{-(-a+yj)^2} \mid dy+ \int^0_{-b} \mid e^{-(a+yj)^2} \mid dy \rightarrow0$$

Thanks a lot (really) for your help,