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Let z=a+ib=re^(i theta) where a, b, thet...

Let `z=a+ib=re^(i theta)` where `a, b, theta in R` and `i=sqrt(-1)` Then `r=sqrt((a^(2)+b^(2)))=|Z|` and `theta=tan^(-1)((b)/(a))=arg(z)` Now `|z|^(2)=a^(2)+b^(2)=(a+ib)(a-ib)=zbar(z) rArr(1)/(2)=(bar(z))/(|z|^(2))` and `|z_(1)z_(2)z_(3)......z_(n)|=|z_(1)||z_(2)||z_(3)|...|z_(n)|` If `|f(z)|=1` ,then `f(z)` is called unimodular. In this case `f(z)` can always be expressed as `f(z)=e^(i alpha), alpha in R` Also `e^(i alpha)+e^(i beta)=e^(i((alpha+beta)/(2)))*2cos((alpha-beta)/(2))` and `e^(i alpha)-e^(i beta)=e^(i((alpha+beta)/(2)))*2i sin((alpha-beta)/(2))` where `alpha, beta in R`
Q:If `Z_(1),Z_(2),Z_(3)` are complex number such that `|Z_(1)|=|Z_(2)|=|Z_(3)|=|Z_(1)+Z_(2)+Z_(3)|=1` , then `|(1)/(Z_(1))+(1)/(Z_(2))+(1)/(Z_(3))|` is

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