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Complex Integrals

  1. Apr 21, 2009 #1
    1. The problem statement, all variables and given/known data

    integral(e^(iz)dz) with bounds from pi to i.
    I need to evaluate using : along the straight line joining the limits, along segments of the coordinate axes joining the limits, and estimate using integral <= L*M and compare the estimates with the values.

    I am totally confused on how to do this. Do I convert z to x+iy and do the bounds change?



    I have another integral with z conjugate about the unit circle. I think I have an idea on this one on converting to x-iy and seeing that this will equal 2ipi.



    I am also asked to evaluate this same integral taken in the positive sense about the square with x=+-1, y=+-1. I know this involves 4 segments, but my problem is finding this 4 segments. I know the answer is 8i, but how to get there?
     
  2. jcsd
  3. Apr 21, 2009 #2
    You can parametrize the path from pi to i, e.g by taking:

    z(t) = pi + (i-pi)t

    The starting point is then at t = 0 and the end point at t = 1.

    integral f(z)dz along the path then means:

    f(z(t))d(z(t)) from t = 0 to t = 1.

    We have dz = (i-pi) dt

    and

    f(z(t)) = exp[i pi - (i pi + 1)t] = -exp[-(i pi + 1)t]
     
  4. Apr 21, 2009 #3
    That makes a lot of sense actually.
     
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