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Question Number 189815 by sonukgindia last updated on 22/Mar/23

	Answered by Frix last updated on 22/Mar/23
	$z=e^(iθ) =cos θ +i sin θ =c+si [for typing less] z^z =(e^(iθ) )^(c+si) =e^(−θs+iθc) =(1/e^(θs) )e^(iθc) z^z =e^(−θsin θ) e^(iθcos θ) ∣z^z ∣=e^(−θsin θ) −π<θ≤π ⇒ mMaximum =1 at θ∈{−π, 0, π} The minimum is harder to find ((d∣z^z ∣)/dθ)=0 −(sin θ +θcos θ)e^(−θsin θ) =0 sin θ +θcos θ =0 Minimum ≈.162073435459 at θ≈±2.02875783811$
	$z = e^{i θ} = \cos θ + i \sin θ = c + s i [for typing less]$ $z^{z} = {(e^{i θ})}^{c + s i} = e^{- θ s + i θ c} = \frac{1}{e^{θ s}} e^{i θ c}$ $z^{z} = e^{- θ \sin θ} e^{i θ \cos θ}$ $∣ z^{z} ∣= e^{- θ \sin θ}$ $- π < θ ⩽ π \Rightarrow mMaximum = 1 at θ \in {- π, 0, π}$ $The minimum is harder to find$ $\frac{d ∣ z^{z} ∣}{d θ} = 0$ $- (\sin θ + θ \cos θ) e^{- θ \sin θ} = 0$ $\sin θ + θ \cos θ = 0$ $Minimum \approx . 162073435459 at θ \approx \pm 2.02875783811$
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