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
$$\mathrm{That}'\mathrm{s}\:\mathrm{what}\:\mathrm{I}\:\mathrm{did}\:\mathrm{before} \\ $$$$\mathrm{It}\:\mathrm{seems}\:\mathrm{you}\:\mathrm{don}'\mathrm{t}\:\mathrm{know}\:\mathrm{much}\:\mathrm{about}\:\mathrm{this}... \\ $$
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
$${f}\left({z}\right)={z}\mathrm{e}^{\mathrm{i}{z}} ={r}\mathrm{e}^{\mathrm{i}\theta} \mathrm{e}^{\mathrm{i}{r}\mathrm{e}^{\mathrm{i}\theta} } = \\ $$$$={r}\mathrm{e}^{\mathrm{i}\theta} \mathrm{e}^{\mathrm{i}{r}\left(\mathrm{cos}\:\theta\:+\mathrm{i}\:\mathrm{sin}\:\theta\right)} = \\ $$$$={r}\mathrm{e}^{\mathrm{i}\theta} \mathrm{e}^{−{r}\mathrm{sin}\:\theta\:+\mathrm{i}{r}\mathrm{cos}\:\theta} = \\ $$$$={r}\mathrm{e}^{−{r}\mathrm{sin}\:\theta\:+\mathrm{i}\left(\theta+{r}\mathrm{cos}\:\theta\right)} = \\ $$$$=\frac{{r}}{\mathrm{e}^{{r}\mathrm{sin}\:\theta} }\mathrm{e}^{\mathrm{i}\left(\theta+{r}\mathrm{cos}\:\theta\right)} = \\ $$$$=\frac{{r}\mathrm{cos}\:\left(\theta+{r}\mathrm{cos}\:\theta\right)}{\mathrm{e}^{{r}\mathrm{sin}\:\theta} }+\mathrm{i}\frac{{r}\mathrm{sin}\:\left(\theta+{r}\mathrm{cos}\:\theta\right)}{\mathrm{e}^{{r}\mathrm{sin}\:\theta} } \\ $$
Commented by Mastermind last updated on 09/Nov/22
$$\mathrm{Thanks}\:\mathrm{man} \\ $$