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Question Number 180277 by Mastermind last updated on 09/Nov/22

$$\mathrm{Express}\mathrm{the}\mathrm{function}\mathrm{f}\left(\mathrm{z}\right)={\mathrm{ze}}^{\mathrm{iz}}\mathrm{into}$$$$\mathrm{cartesian}\mathrm{form}\mathrm{and}\mathrm{separate}\mathrm{it}\mathrm{into}$$$$\mathrm{Real}\mathrm{and}\mathrm{Imaginary}\mathrm{part}.$$$$$$$$\mathrm{M}.\mathrm{m}$$

Commented by Frix last updated on 09/Nov/22

$${\mathrm{That}}^{\prime }\mathrm{s}\mathrm{what}\mathrm{I}\mathrm{did}\mathrm{before}$$$$\mathrm{It}\mathrm{seems}\mathrm{you}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{know}\mathrm{much}\mathrm{about}\mathrm{this}...$$

Commented by Mastermind last updated on 09/Nov/22

$$\mathrm{Smile}$$$$\mathrm{okay},\mathrm{lets}\mathrm{see}\mathrm{in}\mathrm{polar}\mathrm{form}...$$$$\mathrm{when}\mathrm{z}={\mathrm{re}}^{\mathrm{i}\theta }$$

Commented by Frix last updated on 09/Nov/22

$$f\left(z\right)=z{\mathrm{e}}^{\mathrm{i}z}=r{\mathrm{e}}^{\mathrm{i}\theta }{\mathrm{e}}^{\mathrm{i}r{\mathrm{e}}^{\mathrm{i}\theta }}=$$$$=r{\mathrm{e}}^{\mathrm{i}\theta }{\mathrm{e}}^{\mathrm{i}r(\cos \theta +\mathrm{i}\sin \theta )}=$$$$=r{\mathrm{e}}^{\mathrm{i}\theta }{\mathrm{e}}^{-r\sin \theta +\mathrm{i}r\cos \theta }=$$$$=r{\mathrm{e}}^{-r\sin \theta +\mathrm{i}(\theta +r\cos \theta )}=$$$$=\frac{r}{{\mathrm{e}}^{r\sin \theta }}{\mathrm{e}}^{\mathrm{i}(\theta +r\cos \theta )}=$$$$=\frac{r\cos (\theta +r\cos \theta )}{{\mathrm{e}}^{r\sin \theta }}+\mathrm{i}\frac{r\sin (\theta +r\cos \theta )}{{\mathrm{e}}^{r\sin \theta }}$$

Commented by Mastermind last updated on 09/Nov/22

$$\mathrm{Thanks}\mathrm{man}$$

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