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Contour Integral homework

  1. Mar 28, 2010 #1
    s25dts.jpg

    So the length of the contour is L(gamma) = 2.pi

    and so i have http://images.planetmath.org:8080/cache/objects/7138/js/img1.png

    so i need to show max f(z) = e?

    So the maximum of f(z)= e1/z2 in the unit circle centre 0, radius 1 implies that 1/z2 should be maximum, and this is when z2 is its lowest possible value.

    When this happens f(z) cannot equal e unless the lowest possible value of z is 1, but isn't that the highest possible value?

    Any help would be great thanks
     

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    Last edited by a moderator: Apr 24, 2017
  2. jcsd
  3. Mar 28, 2010 #2
    On the unit circle z = exp(i theta). So you need to find an upper limit for

    |exp[exp(-2 i theta)]|

    Hint: Split the inner exponential in its imaginary and real parts.
     
  4. Mar 28, 2010 #3
    so exp[cos(2t) - i.sin(2t)]

    i have an example similar to this stage where i.sin(t) suddenly disappears, so I assume it does that here as well. Could you explain why for me please?

    And then the upper bound of cos(2t) = 1 and so the result follows.
     
  5. Mar 28, 2010 #4
    exp[cos(2t) - i.sin(2t)] = exp[cos(2t)] exp[- i.sin(2t)]

    And then when you take the absolute value, you use that

    |exp(i p)| = 1 for real p.
     
  6. Mar 28, 2010 #5
    ah i didn't know that, thanks
     
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