Complex exponential function

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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

 

Answers and Replies

  • #2
vela
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Let z=x+iy. Suppose you take the limit along the line x=0. What happens?
 
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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 [itex]z=\rho e ^{i\theta} \Rightarrow e^{-z^2}=e^{-\rho^2e^{2i\theta}}[/itex], the magnitude is really dependent on [itex]Re(e^{2i\theta})=\cos 2\theta>0[/itex], and that's where the [itex]|arg(z)|<\pi/4[/itex] from
 
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