simple complex number

• Mar 15th 2010, 05:31 AM
z1llch
simple complex number
if z=x+yi, find all possible values of z such that z^3=1

please enlighten me, no idea where to start.
• Mar 15th 2010, 05:55 AM
HallsofIvy
Quote:

Originally Posted by z1llch
if z=x+yi, find all possible values of z such that z^3=1

please enlighten me, no idea where to start.

It is simplest to write z in "polar form"- [/tex]z= r(cos(\theta)+ i sin(\theta))[/tex] (Engineering notation: $z= r cis(\theta)$ for "cos+ i s sin") which can also be written as $re^{i\theta}$ where "r" is the straight line distance from 0 to x+ iy in the "complex plane" ((0,0) to (x,y)) and [tex]\theta[/itex] is the angle that line makes with the positive real axis (positive x-axis).

Then you can use De Moivres theorem: $[r(cos(\theta)+ i sin(\theta))]^n= r^n (cos(n\theta)+ i sin(n\theta)$ or, in exponential form, $(re^{i\theta}= r^n e^{i n\theta})$.

There, n can be any real number including 1/3: $z^{1/3}= r^{1/3}(cos(\theta/3)+ i sin(\theta/3))$.

Of course, the solutions to $z^3= 1$ are $z= (1)^{1/3}$.

"1" is ON the real-axis to $\theta= 0$ and its distance from 0 is 1 so r= 1: $z^3= 1= 1(cos(0)+ i sin(0))$.

That's obviously true: cos(0)= 1, sin(0)= 0. It is also obvious that $z= 1^{1/3}(cos(0/3)+ i sin(0/3)= 1(cos(0)+ i sin(0))= 1$. Of course you knew that one solution of $z^3= 1$ was z= 1.

Now the good part: Since sine and cosine are periodic with period $2\pi$, $z^3$ can also be written as $1(cos(2\pi)+ i sin(2\pi))$ or as $1(cos(4\pi)+ i sin(4\pi))$ but dividing by 3, so that the sine and cosine are of $\frac{2\pi}{3}$ and $\frac{4\pi}{3}$ gives different results! That's how you can get the two other roots.

(Adding another $2\pi$ would give $6\pi$ and dividing that by 3 gives $2\pi$ again and so not a new solution. If you keep adding " $2\pi$", you cycle though those same three solutions.)