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Can someone explain this equality to me (complex variables)

  1. Apr 7, 2014 #1
    1. The problem statement, all variables and given/known data

    I hate to upload the whole problem, but I am trying to evaluate an indefinite integral, and I can follow the solution until right near the end. The example says that for a point on [itex]C_R[/itex][tex]|e^{-3z}|=e^{-3y}\leq 1[/tex]. I don't understand how they can say this. Below is the question, with a drawing of the region. I have highlighted the step that I do not understand.

    http://media.newschoolers.com/uploads/images/17/00/70/52/45/705245.png [Broken]

    2. Relevant equations

    3. The attempt at a solution

    I might be missing something easy, but I can't see how this is true!
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Apr 7, 2014 #2

    LCKurtz

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    ##e^{3iz}=e^{3i(x+iy)}=e^{-3y+3ix}=e^{-3y}e^{3ix}## and ##|e^{i3x}|=1##.
     
  4. Apr 7, 2014 #3
    exp(3iz) = exp(3ix - 3y) = exp(-3y) exp(3ix)
    The magnitude of this complex number is |exp(-3y)| times 1, because exp(3ix) = cos(3x) + i sin(3x), and |exp(3ix)| is the sum of a squared cosine and a squared sine of the same argument. And then of course |exp(-3y)| = exp(-3y) because e-to-the-anything is always positive.

    Why is exp(-3y) <= 1? Because exp(0) = 1, and on the given semicircular path, y is non-negative.
     
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