Find the pricipal value of z=(1−i)^i


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Question Number 31335 by NECx last updated on 06/Mar/18

$F i n d t h e p r i c i p a l v a l u e o f$ $z = {(1 - i)}^{i}$
Commented by abdo imad last updated on 06/Mar/18

$w e h a v e z = e^{i l n (1 - i)} a n d l n h e r e m e s n d t h e c o m p l e x e l o g$ $b u t 1 - i = \sqrt{2} (\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}) = e^{- i \frac{π}{4}} \Rightarrow l n (1 - i) = l n (\sqrt{2}) - \frac{i π}{4} \Rightarrow$ $i l n (1 - i) = i l n (\sqrt{2}) + \frac{π}{4} \Rightarrow e^{i l n (1 - i)} = e^{\frac{π}{4}} (c o s (l n (\sqrt{2}) + i s i n (l n (\sqrt{2}))) \Rightarrow$ ${(1 - i)}^{i} = e^{\frac{π}{4}} c o s (l n (\sqrt{2})) + i e^{\frac{π}{4}} s i n (l n (\sqrt{2})) .$
Commented by abdo imad last updated on 06/Mar/18

$∣ {(1 - i)}^{i} ∣= e^{\frac{π}{4}} .$
Commented by NECx last updated on 06/Mar/18

$t h a n k y o u s o m u c h .$
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