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Question Number 184731 by Mastermind last updated on 11/Jan/23

$$\mathrm{Express}\mathrm{this}\mathrm{function}\mathrm{in}\mathrm{both}\mathrm{its}$$$$\mathrm{Cartesian}\mathrm{and}\mathrm{polar}\mathrm{form}$$$$\mathrm{f}\left(\mathrm{z}\right)={\mathrm{ze}}^{\mathrm{iz}}.$$$$$$$$$$$$\mathrm{Help}!$$

Answered by MJS_new last updated on 11/Jan/23

$$1^{\mathrm{st}}\mathrm{possibility}$$$$z=a+b\mathrm{i}=\sqrt{a^2+b^2}{\mathrm{e}}^{\mathrm{i}\arctan \frac{b}{a}}$$$$\Rightarrow$$$$f\left(z\right)=\sqrt{a^2+b^2}{\mathrm{e}}^{\mathrm{i}\arctan \frac{b}{a}}{\mathrm{e}}^{-b+a\mathrm{i}}=$$$$=\frac{\sqrt{a^2+b^2}}{{\mathrm{e}}^b}{\mathrm{e}}^{\mathrm{i}(a+\arctan \frac{b}{a})}=$$$$=\frac{\sqrt{a^2+b^2}}{{\mathrm{e}}^b}(\cos (a+\arctan \frac{b}{a})+\mathrm{i}\sin (a+\arctan \frac{b}{a}))$$$$$$$$2^{\mathrm{nd}}\mathrm{possibility}$$$$z=r{\mathrm{e}}^{\mathrm{i}\theta }=r\cos \theta +\mathrm{i}r\sin \theta$$$$\Rightarrow$$$$f\left(z\right)=r{\mathrm{e}}^{\mathrm{i}\theta }{\mathrm{e}}^{-r\sin \theta +\mathrm{i}r\cos \theta }=$$$$=\frac{r}{{\mathrm{e}}^{r\sin \theta }}{\mathrm{e}}^{\mathrm{i}(\theta +r\cos \theta )}=$$$$=\frac{r}{{\mathrm{e}}^{r\sin \theta }}(\cos (\theta +r\cos \theta )+\mathrm{i}\sin (\theta +r\cos \theta ))$$

Commented by Mastermind last updated on 11/Jan/23

$${\mathrm{I}}^{\prime }\mathrm{m}\mathrm{just}\mathrm{seeing}\mathrm{your}\mathrm{solution}$$$$\mathrm{Ok}\mathrm{now},\mathrm{Assuming}\mathrm{we}\mathrm{were}\mathrm{given}$$$$\mathrm{the}\mathrm{value}\mathrm{of}\theta .$$$$\mathrm{how}\mathrm{we}\mathrm{want}\mathrm{to}\mathrm{do}\mathrm{ordinary}\theta ?$$

Commented by MJS_new last updated on 11/Jan/23

$$\mathrm{sorry}\mathrm{but}\mathrm{I}\mathrm{do}\mathrm{not}\mathrm{understand}:\mathrm{what}\mathrm{do}\mathrm{you}$$$$\mathrm{mean}\mathrm{with}‘‘\mathrm{ordinary}{\theta }^{″}?$$

Commented by MJS_new last updated on 19/Jan/23

$$\mathrm{well},\mathrm{simply}\mathrm{insert}\mathrm{the}\mathrm{value}\mathrm{of}\theta$$

Commented by Mastermind last updated on 19/Jan/23

$$\mathrm{I}\mathrm{meant}\mathrm{if}\mathrm{the}\mathrm{value}\mathrm{of}\theta \mathrm{given},\mathrm{how}\mathrm{are}$$$$\mathrm{going}\mathrm{to}\mathrm{do}\mathrm{it}\mathrm{in}\mathrm{the}\mathrm{answer}\mathrm{you}\mathrm{got}$$

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