Question 7.5.5: Consider the complex number z = 0.5 + 0.8i. Represent the po......

To study the powers, write z in polar form:

z = r(\cos θ + i \sin θ ).

Here

r =\sqrt{0.5^2 + 0.8^2} ≈ 0.943

and

θ = \arctan\frac{0.8}{0.5}≈ 58^◦.

We have

z^n = r^n \cos(nθ ) + i \sin(nθ )).

The vector representation of z^{n+1} is a little shorter than that of z^n (by about 5.7%), and z^{n+1} makes an angle θ ≈ 58^◦ with z^n. If we connect the tips of consecutive vectors, we see a trajectory that spirals in toward the origin, as shown in Figure 11. Note that \lim\limits_{n \to ∞} z^n=0, since r = |z| < 1.