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

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The function f(t) = Re(t^i) and the parametric equations x(t) = Re(t^i) and y(t) = Im(t^i) exhibit infinitely many turning points between t=0 and t=1. Graphical analysis using Mathematica supports the conclusion of infinite turning points and revolutions traced out by the parametric function, which resembles a unit circle. The discussion emphasizes that this behavior persists for any positive real number a in the domain 0 < t < a. Participants are encouraged to share insights or further reading on the topic. The exploration of these functions reveals complex behavior in their graphical representations.
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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.
 
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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. :)
 
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