# Complex exponential function

1. Apr 12, 2013

### liyz06

1. The problem statement, all variables and given/known data

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...

2. Relevant equations

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

3. The attempt at a solution

2. Apr 12, 2013

### vela

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

3. Apr 12, 2013

### liyz06

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

Last edited: Apr 12, 2013