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Calculate the following contour integrals sing suitable parametr

  1. May 3, 2012 #1
    Calculate the following contour integrals using suitable parameterisations

    1. The problem statement, all variables and given/known data

    1)##\oint \frac{1}{z-z_0} dz## where C is the circle ##z_0## and radius r>0 oriented CCW and ##k\ge0##

    2) ##\int_c |z|^2 dz## where C is the straight line from 1+i to -1

    3. Relevant equations

    ##\oint f(z)dz=\int_a^b f(z(t))*z'(t)dt##

    3. The attempt at a solution

    1) let z(t)=e^it for ##0\le t \le1## and ##z_0=0## since it is centred at ##(0,0)##

    We have ##\displaystyle \int_c \frac{1}{z^k} dz=\int_0^{2\pi} \frac{1}{e^{ikt}} i e^{it} dt=\frac{i}{(1-k)}e^{it}|_0^{2\pi}##..?


    2) ##\int_c |z|^2 dz## where C is the straight line from 1+i to -1

    let ##z(t)=(1-t)(1+i)+t(0)=1-t+i(1-t)## for ##0 \le t \le 1##

    ##z'(t)=-1-i##

    ##|z(t)|^2=\sqrt((1-t)^2+(1-t)^2)=2t^2-4t+2##

    Therefore ##\int_c |z|^2 dz=\int_0^1 (2t^2-4t+2)(-1-i)dt##...?

    How am I doing so far on both?

    Thanks
     
    Last edited: May 3, 2012
  2. jcsd
  3. May 3, 2012 #2
    Re: Calculate the following contour integrals using suitable parameterisations

    Your reasoning isn't consistent here. What is k in the original problem? What is ##z_0##? Is the circle centered at ##z_0## with positive radius r? I can't help you until that is cleared up.

    Your formula for z(t) is wrong. z(1) equals 0, not -1. You meant to write ##z(t) = (1-t)(1+i) + t(-1)##

    Your method seems correct after that step. Also, ##|z|^2 = z \bar{z}##, it miught be easier to do that, might not.
     
  4. May 4, 2012 #3
    Re: Calculate the following contour integrals using suitable parameterisations

    C is the circle with centre z_0 and radius r>0 orientated CCW. I left out the k, it should read

    ##\oint \frac{1}{(z-z_0)^k} dz##

    I dont know how to handle the z_0...?


    Sorry, thats a mistake on my part. Will review..
     
  5. May 4, 2012 #4
    Re: Calculate the following contour integrals using suitable parameterisations

    Okay! So what is your equation equation for a circle centered at ##z_0## with positive radius? Then use the contour integral formula. You should come out with an answer very similar to what your had in your first post.

    ##\displaystyle \int_c \frac{1}{(z - z_0)^k} dz=\int_0^{2\pi} \frac{1}{(z_0 + re^{it} - z_0)^k} (ir e^{it}) dt= \dots##

    Make sure your anti-derivitive is correct, your anti-derivative in your first post was wrong.

    Also, is k any real number greater than or equal to zero or any integer greater than or equal to zero? I'm inclined to believe the latter but the choice of what k is will determine what the final answer is.
     
    Last edited: May 4, 2012
  6. May 4, 2012 #5
    Re: Calculate the following contour integrals using suitable parameterisations

    Yes it is the latter. I see you have taken ##z=re^{it}+z_0## SO the integral should work out to be

    ##\displaystyle \frac{ir^2}{(1-k)}e^{it(1-k)} |_0^{2\pi}##...?
     
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