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Contour integral (from complex analysis)

  1. Mar 7, 2010 #1
    1. The problem statement, all variables and given/known data

    let g denote the elliptic arc parametrized by z(t) = 2cost + 3isint, for t between 0 and pi/2 (inclusive).

    Evaluate the integral of f(z) = z[sin(pi*z^2) - cos(pi*z^2)] over g.



    2. Relevant equations

    If g is determined by the function z mapping from [a,b] to C and f maps from g to C, then the integral of f over g is defined as the integral (from a to b) of f of z(t) times z'(t).

    (sorry for writing the equations out in words, I don't have any formatting software)



    3. The attempt at a solution

    I started by finding z'(t) = -2sint + 3icost and attempting to find f(z(t)), but I got a really complicated function and at that point I figured I must be going about it the wrong way.

    I tried to find an identity that would allow me to simplifiy f(z) but I couldn't find anything.

    At this point I really have no idea how to proceed.
     
  2. jcsd
  3. Mar 7, 2010 #2

    vela

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    If f(z) is analytic, what do you know about its integral along a contour between two points?
     
  4. Mar 7, 2010 #3
    hmm...I don't know. It doesn't say so in the problem. Is that something I should be able to recognize?
     
  5. Mar 7, 2010 #4

    vela

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    Yes, or at least, it's something they want you to learn. I imagine that's the point of this problem.
     
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