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Complex exponential function

  1. Apr 12, 2013 #1
    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. jcsd
  3. Apr 12, 2013 #2

    vela

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    Let z=x+iy. Suppose you take the limit along the line x=0. What happens?
     
  4. Apr 12, 2013 #3
    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
     
    Last edited: Apr 12, 2013
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