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Thread: Complex analysis

  1. #1
    Kai is offline
    Junior Member
    Apr 2008

    Complex analysis

    Hi, need some help to clear some doubts, have test on monday

    Find all the values of :
    $\displaystyle (1+\sqrt3i)^{1/3}$

    I got : $\displaystyle z=2^{1/3}*e^{i({2k\pi}+\pi/3})$ ,k = 0,1,2

    $\displaystyle z_{0}=2^{1/3}*e^{i\pi/9}$, $\displaystyle z_{1}=2^{1/3}*e^{i7\pi/9}$,$\displaystyle z_{2}=2^{1/3}*e^{-i5\pi/9}$

    Can somebody chek my answers, coz am not sure they are good, btw i see in my notes that the nth roots of unity forms a regular polygon, in our case it should form an equilateral triangle ? or is it only when $\displaystyle z^x=1$ that it holds ??
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  2. #2
    Super Member
    Aug 2008
    Yea, that's right. The central issue is $\displaystyle \frac{\Theta}{n}+\frac{2k\pi}{n}$ in the expression $\displaystyle z^{1/n}=r^{1/n}e^{i/n(\Theta+2k\pi)}$.

    So you're breaking up the circle into $\displaystyle \frac{2k\pi}{n}$ sectors of equal size starting at the point $\displaystyle \Theta/n$ with a root at each terminal point of each sector so of course they're going to be equally spaced around a circle of radius $\displaystyle Abs(z)^{1/n}$ and therefore at the points of a regular polygon.

    Here's an idea. Write a Mathematica application which illustrates the roots of $\displaystyle z^{1/n}$ as red points in the complex plane for a variable sized z for n running from 2 to 10. Sort of like this:

      pts = N[w /. Solve[r*Exp[I*tval] == w^n, 
           w]]; points = 
        (Point[{Re[#1], Im[#1]}] & ) /@ pts; 
       Show[Graphics[{Red, PointSize[0.02], 
          points}], PlotRange -> 
         {{-3, 3}, {-3, 3}}, Axes -> True], 
      {r, 1, 5}, {tval, 0, Pi/2, 0.1}, 
      {n, 2, 10, 1}]
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