Prove that if z ∈ C and Re(z^k) ≥ 0 for all k = 1, 2, 3, . . ., then z ∈ [0,∞).

Question

anonymous · Answer

@hartnn Are you able to help?

zarkon · Answer

use the polar form of a complex number

anonymous · Answer

so $z^k=r^k(cos(k\theta) +isin(k\theta)$ and thus $Re(z^k)=r^kcos(k\theta) \ge 0 $. but i am not entirely sure what the question is asking, specifically the interval given. What actually needs to be proved?

zarkon · Answer

prove that that $\theta=0$

anonymous · Answer

oh ok that makes more sense to me now as the sine term will vanish. How would i go about proving $\theta = 0 $

zarkon · Answer

Assume $\theta\ne0$ and derive a contadiction

zarkon · Answer

you will also have to take care of the times when $cos(\theta)=0$

ie $\theta=\pm\dfrac{\pi}{2}$

though those are easy to resolve.

anonymous · Answer

can you help me with the proof, im not sure how to do it

zarkon · Answer

show there is some k such that $\cos(k\theta)<0$

anonymous · Answer

i'm stuck i dont know how to start the proof

anonymous · Answer

@phi you able to help?

anonymous · Answer

@thomaster Are you able to help?

anonymous · Answer

@goformit100

goformit100 · Answer

Ya say?

anonymous · Answer

are you able to help me with this proof?

goformit100 · Answer

Sorry Sir I don't know it's proof

anonymous · Answer

ok no worries

zarkon · Answer

if i give you much more...i might as well do the entire problem