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Finding an upper bound for a contour integral (Complex)

  1. Aug 20, 2016 #1
    C1 1. The problem statement, all variables and given/known data:
    Using the ML inequality, I have to find an upper bound for the contour integral of \int e^2z - z^2 \, dz
    where the contour C = C1 + C2.

    C1 is the circular arc from point A(sqrt(3)/2, 1/2) to B(1/2, sqrt(3)/2) and C2 is the line segment from the origin to B.


    2. Relevant equations


    3. The attempt at a solution

    I figured I have to find M1 and M2, and also L1 and L2 given the two different contours. Along C1, I have found the length to be pi/6 by integrating a parameterized equation of the unit circle from pi/6 to pi/3. I have also found the length along C2 to be 1 (no surprises here)

    However, I've hit a bump when trying to find the upper bound for e^2z - z^2.
    I figured I should use the triangle inequality i.e. |e^2z - z^2|
    <= |e^2z| + |-z^2|
    = |e^2z| + |z^2|
    = |e^2z| + |z|^2
    = |e^2z| + 1 (since the length of the line segment is one).

    I'm kind of stuck when it comes to the e^2z. It's the same problem for finding the upper bound on C2 as well.

    I'm sorry for not using Latex. I tried formatting it but it doesn't come out as intended in the Preview, so I decided not to take the risk of fudging it up and making it look confusing. I'm quite sleepy, sorry.
     
  2. jcsd
  3. Aug 20, 2016 #2

    pasmith

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    Homework Helper

    If [itex]z = x + iy[/itex] then [itex]e^z = e^x (\cos y + i \sin y)[/itex]...
     
  4. Aug 20, 2016 #3
    Thank you so much, I've got it :)
     
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