Express-this-f-z-2z-i-z-i-in-polar-form-where-z-re-i-polar-form-

Question Number 180285 by Mastermind last updated on 09/Nov/22

Express this f(z)=((2z+i)/(z+i)) in polar form where z=re^(iθ) (polar form)

$$\mathrm{Express}\:\mathrm{this}\:\mathrm{f}\left(\mathrm{z}\right)=\frac{\mathrm{2z}+\mathrm{i}}{\mathrm{z}+\mathrm{i}}\:\mathrm{in}\:\mathrm{polar}\:\mathrm{form} \\ $$$$\mathrm{where}\:\mathrm{z}=\mathrm{re}^{\mathrm{i}\theta} \:\left(\mathrm{polar}\:\mathrm{form}\right) \\ $$$$ \\ $$

Commented by Frix last updated on 09/Nov/22

do some work too it′s similar to the last one, we need the Cartesian form to get there: f(z)=((2r^2 +3rsin θ +1)/(r^2 +2rsin θ +1))−i((rcos θ)/(r^2 +2rsin θ +1))= =(√((4r^2 +4rsin θ +1)/(r^2 +2rsin θ +1)))e^(−i arctan ((rcos θ)/(2r^2 +3rsin θ +1)))

$$\mathrm{do}\:\mathrm{some}\:\mathrm{work}\:\mathrm{too} \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{similar}\:\mathrm{to}\:\mathrm{the}\:\mathrm{last}\:\mathrm{one},\:\mathrm{we}\:\mathrm{need}\:\mathrm{the} \\ $$$$\mathrm{Cartesian}\:\mathrm{form}\:\mathrm{to}\:\mathrm{get}\:\mathrm{there}: \\ $$$${f}\left({z}\right)=\frac{\mathrm{2}{r}^{\mathrm{2}} +\mathrm{3}{r}\mathrm{sin}\:\theta\:+\mathrm{1}}{{r}^{\mathrm{2}} +\mathrm{2}{r}\mathrm{sin}\:\theta\:+\mathrm{1}}−\mathrm{i}\frac{{r}\mathrm{cos}\:\theta}{{r}^{\mathrm{2}} +\mathrm{2}{r}\mathrm{sin}\:\theta\:+\mathrm{1}}= \\ $$$$=\sqrt{\frac{\mathrm{4}{r}^{\mathrm{2}} +\mathrm{4}{r}\mathrm{sin}\:\theta\:+\mathrm{1}}{{r}^{\mathrm{2}} +\mathrm{2}{r}\mathrm{sin}\:\theta\:+\mathrm{1}}}\mathrm{e}^{−\mathrm{i}\:\mathrm{arctan}\:\frac{{r}\mathrm{cos}\:\theta}{\mathrm{2}{r}^{\mathrm{2}} +\mathrm{3}{r}\mathrm{sin}\:\theta\:+\mathrm{1}}} \\ $$

Commented by Mastermind last updated on 10/Nov/22

$$\mathrm{ok} \\ $$

Commented by Mastermind last updated on 10/Nov/22

$$\mathrm{The}\:\mathrm{thing}\:\mathrm{is}\:\mathrm{kind}\:\mathrm{of}\:\mathrm{complex}\:\mathrm{ooo} \\ $$

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