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Graphing a function under a complex mapping

  1. Jan 26, 2017 #1
    1. The problem statement, all variables and given/known data
    Illustrate the mapping of [itex] f(z)=z+\frac{1}{z} [/itex]
    for a parametric line.
    3. The attempt at a solution
    the equation for a parametric line is [itex] z(t)=z_0(1-t)+z_1(t) [/itex]
    so I plug z(t) in for z in f(z), but I dont get an obvious expression on how to graph it,
    I tried manipulating it,
    Was also wondering if I should represent [itex] f(z)=\frac{(z-i)(z+i)}{z} [/itex]
     
    Last edited by a moderator: Jan 26, 2017
  2. jcsd
  3. Jan 26, 2017 #2

    Ray Vickson

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    Pick some numerical values ##z_1 = x_1 + i y_1## and ##z_2 = x_2 + i y_2##, then look at ##f(z) = z + 1/z## at ##z = z_1 (1-t) + z_2\, t##:
    $$\begin{array}{rcl}f(z) &=&z_1 (1-t) + z_2\, t + \frac{1}{z_1 (1-t) + z_2\, t}\\
    &=& \displaystyle (x_1+ i y_1)(1-t) + (x_2 + iy_2) t + \frac{1}{ (x_1+ i y_1)(1-t) + (x_2 + iy_2) t }
    \end{array}$$
    After some algebra this will have the form ##A(t) + i B(t)## for some functions ##A, B##, so you get a parametric curve of the form ##x = A(t)##, ##y = B(t)## to plot.
     
    Last edited: Jan 27, 2017
  4. Jan 27, 2017 #3
    ok thanks, I also need to do it for a circle, For a circle the
    equtation is [itex] z(t)=re^{it} [/itex]
    so If I plug this into f(z) I get , and im assuming r=1 for this [itex] e^{it}+e^{-it} [/itex]
    which is 2cos(t), so then Ijust graph 2*cos(t) as my answer.
     
  5. Jan 27, 2017 #4

    haruspex

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    That does not follow the recipe Ray gave you. He explained that you should plot y against x, not f against t.
    Also, it would be better to avoid assuming r=1.
     
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