If $z$ and $w$ are two non zero complex numbers such that $|zw| =1$ and $\arg z - \arg w = \pi/2$ then conjugate of $(zw)$ =?
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2Does "(z*w)" = $zw$ or $\bar z w$ or perhaps something else? – Robert Lewis Apr 27 '14 at 07:26
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It is (zw) bar [ whole bar] – user142778 Apr 27 '14 at 07:31
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1You mean as in $\overline{zw}$? – Robert Lewis Apr 27 '14 at 07:45
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1yes it is conjugate of whole thing – user142778 Apr 27 '14 at 07:47
2 Answers
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Hint. Write $z$ and $w$ in polar form, $$z=re^{i\alpha}\quad\hbox{and}\quad w=se^{i\beta}\ .$$ Then
- in terms of $r,s$ we have $|zw|=\cdots$
- in terms of $\alpha,\beta$ we have ${\rm Arg}(z)-{\rm Arg}(w)=\cdots$
- and so $\overline{zw}=\cdots$
David
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Let, $$z=|z|e^{i\arg z},\quad w=|w|e^{i\arg w}$$ Hence, $\overline{zw}=|zw|e^{-i(\arg z +\arg w)}=e^{-i(\pi/2+2\arg w)}=-ie^{-i2\arg w}$. Hope this helps.
Samrat Mukhopadhyay
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1Then the question must be asking to find $z\bar{w}$. Just evaluate this the same way I did in my answer, you'll see the answer will be $-i$. – Samrat Mukhopadhyay Apr 27 '14 at 07:57