Infinitely Many Turning Points: A Study of the Functions Re(t^i) and Im(t^i)

[ben297]

Considering the function

f(t) = Re(t^i) where i is the imaginary unit

How many turning points exist between t=0 and t=1?

Similarly, considering the parametric function

x(t) = Re(t^i)
y(t) = Im(t^i)

(which appears to trace out a unit circle), how many revolutions does this make between t=0 and t=1?

I cannot answer either of these questions with any certainty. However, I've been graphing them using Mathematica, and based on that, the answer to both questions appears to be infinitely many.

I'd appreciate any comments or explanations, or any recommended reading.

 

[ben297]

I was able to prove that over the domain

0 < t < a

Re(t^i) and Im(t^i) both have infinitely many turning points. a can be any positive real number of any magnitude. :)