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2 complex analysis problem

  • #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
 

Answers and Replies

  • #2
1,796
53
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)
 
  • #3
1,101
3
For the second one, use the http://en.wikipedia.org/wiki/Estimation_lemma" [Broken].
 
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