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Homework Help: 2 complex analysis problem

  1. Nov 6, 2011 #1
    1.Evaluate ∫C Im(z − i)dz, where C is the contour consisting of the circular arc along |z| = 1 from z = 1 to z = i and the line segment from z = i to z = −1.

    2. Suppose that C is the circle |z| = 4 traversed once. Show that
    §C (ez/(z+1)) dz ≤ 8∏e4/3

    For question 1, should i let z= x+yi to solve the question?
    and it said the Im part so i just need to consider the "yi"?

    i tried but really have no idea on these 2 questions..
    Please help
    Thanks
     
  2. jcsd
  3. Nov 7, 2011 #2
    Yes, let z=x+iy and then integrate iy-i over z=e^(it) as t goes from 0 to pi/2. So wouldn't that be:

    [tex]\int_0^{\pi/2} Im(z-i)dz,\quad z=e^{it}[/tex]

    You can convert that to all in t right? dz=ie^(it)dt=i(cos(t)+isin(t))dt and won't y be just sin(t)?

    For the contour from i to -1, need to do that one in terms of z=x(t)+iy(t). Isn't that line just y=x+1 as x goes from 0 to -1? So suppose I let x=x(t)=t, then y(t) is? Now substitute all that into the integral:

    [tex]\int_0^{-1} (iy-i)dz,\quad z=x(t)+iy(t)[/tex]

    with dz=x'(t)+iy'(t)
     
  4. Nov 7, 2011 #3
    For the second one, use the http://en.wikipedia.org/wiki/Estimation_lemma" [Broken].
     
    Last edited by a moderator: May 5, 2017
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