Complex Integration: Contour Evaluation and Estimation Lemma

[nickolas2730]

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 (}e^z/(z+1)) dz ≤ 8∏e⁴/3

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

i tried but really have no idea on these 2 questions..
Please help
Thanks

 

[jackmell]

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:

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

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:

$$\int_0^{-1} (iy-i)dz,\quad z=x(t)+iy(t)$$

with dz=x'(t)+iy'(t)

 

[snipez90]

For the second one, use the http://en.wikipedia.org/wiki/Estimation_lemma" .