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Plots in an Argand Diagram

  1. May 23, 2014 #1
    1. The problem statement, all variables and given/known data

    Find the locus of points which satisfy

    2π*|z −1| = Arg(z − 1) where |z −1| ≤ 2.

    2. Relevant equations

    n/a

    3. The attempt at a solution

    I know that |z-1| ≤ 2 is the 'inside' bits of a circle center (1,0) with a radius 2

    After that I get confused surely with |z-1|≤2 then Arg(z-1) has to be between 0 and 4π... but then that's all space?

    Thanks in advance
     
  2. jcsd
  3. May 23, 2014 #2
    Since everything is with reference to ##1+0i##, it would be good idea to shift the origin here. Shifting the origin, your equations transform to:
    $$2\pi |z|=\arg(z)$$
    $$|z|\leq 2$$
    Use the substitution ##z=re^{i\theta}## in the first equation, do you see where that leads to?
     
  4. May 23, 2014 #3
    So that means that ##2\pi r=\theta## which is a spiral beginning at (0,0) so to get the answer is it just
    $$2\pi (r-1)=\theta$$
    Thanks!
     
  5. May 23, 2014 #4
    Yes.
    Well, no. I don't think that transformation is correct. Once you plot the graph, you need to move everything by 1 unit towards right.

    Look at the plots of ##2\pi r=\theta## and ##2\pi (r-1)=\theta##.
     
  6. May 24, 2014 #5
    Ah ok got you now :) thank you!
     
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