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Question Number 152203 by peter frank last updated on 26/Aug/21

$$\mathrm{If}\mathrm{x}\mathrm{is}\mathrm{real}\mathrm{show}\mathrm{that}$$$${(2+\mathrm{i})}^{(1+3\mathrm{i})\mathrm{x}}+{(2-\mathrm{i})}^{(1-3\mathrm{i})\mathrm{x}}$$$$\mathrm{is}\mathrm{also}\mathrm{real}$$

Commented by MJS_new last updated on 26/Aug/21

$${\left({\mathrm{e}}^{\theta \mathrm{i}}r\right)}^{a+b\mathrm{i}}+{\left({\mathrm{e}}^{-\theta \mathrm{i}}r\right)}^{a-b\mathrm{i}}=$$$$={\mathrm{e}}^{-\theta b+\theta a\mathrm{i}}{r}^{a+b\mathrm{i}}+{\mathrm{e}}^{-\theta b-\theta a\mathrm{i}}{r}^{a-b\mathrm{i}}=$$$$=\left({\mathrm{e}}^{\theta a\mathrm{i}}{r}^{b\mathrm{i}}\right)\left({\mathrm{e}}^{-\theta b}{r}^a\right)+\left({\mathrm{e}}^{-\theta a\mathrm{i}}{r}^{-b\mathrm{i}}\right)\left({\mathrm{e}}^{-\theta b}{r}^a\right)$$$$\mathrm{easy}\mathrm{to}\mathrm{see}\mathrm{the}\mathrm{shape}\mathrm{of}\mathrm{these}\mathrm{is}$$$${\mathrm{e}}^{\mathrm{i}\varphi }s+{\mathrm{e}}^{-\mathrm{i}\varphi }s=2s\cos \varphi$$$$\mathrm{which}\mathrm{obviously}\mathrm{is}\mathrm{real}$$

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