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Conformal mapping w=1/z - question.

  1. Nov 20, 2013 #1
    Hi,
    1. The problem statement, all variables and given/known data
    I'd like to show that the mapping w=u+iv=1/z tranforms the line x=b in the z plane into a circle with radius 1/2b and center at u=1/2b


    2. Relevant equations



    3. The attempt at a solution
    z*w=1=(b+iy)(u+iv)
    → 1=|(bu-yv)+i(bv+yu)|
    → u2+v2=1/(b2+y2)
    Now, a circle with radius 1/2b and center at u=1/2b in the w plane would have the following form:
    (u-1/2b)2+v2=1/4b2
    → u2+v2=u/b
    My problem is now explicitely showing that
    1/(b2+y2)=u/b
    Which I am unable to do :(.
    Any advice? (By the way, I have tried using polar coordinates too, to no avail. I happen to prefer this approach.)
     
  2. jcsd
  3. Nov 20, 2013 #2

    Dick

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    Science Advisor
    Homework Helper

    I'm not sure where you are really going there. If w=1/z and z=b+iy and w=u+iv, then w=1/(b+iy). Split that into real and imaginary parts to get your u and v.
     
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