Roots of Complex Numbers

Builing on the polar form of a complex number z=|z|Exp[i theta], and properties of the exponential function, one can find roots of complex number.

Sqrt[z]=Sqrt[ |z|Exp[i theta] ]=Sqrt[ |z| ] Exp[i theta/2],
remember, |z| is a REAL number.
In general, z^(1/n)=|z|^(1/n) Exp[i theta/n]
Let's try it.

In[30]:=

  z=3+4 I
  Sqrt[z]

Out[30]=

  3 + 4 I

Out[31]=

  2 + I

In[32]:=

  Sqrt[Abs[z]] Exp[I Arg[z]/2]
  N[Sqrt[Abs[z]] Exp[I Arg[z]/2]]

Out[32]=

           I/2 ArcTan[3, 4]
  Sqrt[5] E

Out[33]=

  2. + 1. I

Shouldn't there be two square roots? If s is a square root, isn't -s also?

In[34]:=

  z=3+4 I
  s=-Sqrt[z]

Out[34]=

  3 + 4 I

Out[35]=

  -2 - I

In[36]:=

s^2

Out[36]=

  3 + 4 I

Look at the plot of the two roots. Do you notice anything?

In[37]:=

  p1=ListPlot[{{2,1},{-2,-1}}, PlotStyle->{RGBColor[1,0,0],PointSize[.02]}]

Out[38]=

  -Graphics-

Look again.

In[39]:=

  p2=PolarPlot[Sqrt[Abs[z]],{t, 0, 2 Pi},DisplayFunction->Identity];
  Show[p1,p2, DisplayFunction->$DisplayFunction,AspectRatio->Automatic]

Out[40]=

  -Graphics-

The two square roots are Pi away from each other on the circle of radius Sqrt[ |z| ].

Let's look at cube roots

In[41]:=

  z
  c1=N[z^(1/3)]

Out[41]=

  3 + 4 I

Out[42]=

  1.62894 + 0.520175 I

There should be three cube roots. The other two are (more on this later):

In[43]:=

  c2=N[Abs[z]^(1/3) Exp[I(Arg[z]/3+(2 Pi/3))]]
  c3=N[Abs[z]^(1/3) Exp[I(Arg[z]/3+(4 Pi/3))]]

Out[43]=

  -1.26495 + 1.15061 I

Out[44]=

  -0.363984 - 1.67079 I

Don't believe me?

In[45]:=

  c2^3
  c3^3

Out[45]=

  3. + 4. I

Out[46]=

  3. + 4. I

In[47]:=

  p3=ListPlot[{{1.62894,.520175},{-1.26495,1.15061},
    {-.363984,-1.67079}},DisplayFunction->Identity, 
    PlotStyle->{RGBColor[1,0,0],PointSize[.02]}];
  p4=PolarPlot[Abs[z]^(1/3),{t,0, 2 Pi},DisplayFunction->Identity];
  Show[p3,p4,DisplayFunction->$DisplayFunction, 
  AspectRatio->Automatic]

Out[48]=

  -Graphics-

The cube roots are evenly spaced around the circle of radius |z|^(1/3).

In general, the nth roots are evenly spaced around the circle of radius |z|^(1/n), starting with the "first", or principal root. Putting this together with the polar form, one can also see that the n nth roots of z are given by
|z|^(1/n) Exp[i (theta/n + 2kPi/n)], k=0, 1, 2, 3, . . ., n-1

Exercise: Verify using Euler's equation that the n numbers given above are indeed nth roots of z.

Up to Complex Numbers