Why Does exp(-z^2) Approach Zero in Certain Sectors?

[liyz06]

Homework Statement

Reading Hinch's book, there is a statement as follows:

... z need to be kept in the sector where exp(-z^2) ->0 as z -> infinity. Thus it's applicable to the sector |arg z|<pi/4...

Homework Equations

Why is this true and what is the limiting behavior of exp(x) for x in different sectors of the complex plane?

The Attempt at a Solution

 

[vela]

Let z=x+iy. Suppose you take the limit along the line x=0. What happens?

 

[liyz06]

Let z=x+iy. Suppose you take the limit along the line x=0. What happens?

I get it now, use polar coordinate then it's $z=\rho e ^{i\theta} \Rightarrow e^{-z^2}=e^{-\rho^2e^{2i\theta}}$, the magnitude is really dependent on $Re(e^{2i\theta})=\cos 2\theta>0$, and that's where the $|arg(z)|<\pi/4$ from