What Is the Infinite Set of the Orbit of z in Complex Analysis?

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Homework Statement



I can't seem to find any books on this or website to learn from. Has anyone even heard of orbit of z besides the in the Mandelbrot set

I am trying to figure out the orbit of z when x= 1 radian and z = cos(x)+isin(x)
I realize that it orbits the unit circle, but I would think that it would take 720 times to orbit back to its original spot. The books answer states the orbit O(z) is an infinite set. Why?

Thanks for the help
 
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The orbit of z under what action? What method is being used to produce new points from old ones?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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