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Question Number 187359 by Humble last updated on 16/Feb/23

$$$$$$whatarethetwocomplexsolutionto$$$$X^{-x}+{(-X)}^x=0inadditionto\pm 1?$$

Answered by Frix last updated on 16/Feb/23

$$x=r\mathrm{i};\mathrm{let}r>0\left[\mathrm{due}\mathrm{to}\mathrm{symmetry}\right]$$$${\left(r\mathrm{i}\right)}^{-r\mathrm{i}}+{(-r\mathrm{i})}^{r\mathrm{i}}=0$$$${2\mathrm{e}}^{\frac{\pi r}{2}}\cos \left(r\ln r\right)=0$$$$\cos \left(r\ln r\right)=0$$$$r\ln r=(n-\frac{1}{2})\pi ;n\in \mathbb{Z}$$$$\mathrm{Since}\mathrm{the}\mathrm{minimum}\mathrm{of}r\ln r\mathrm{is}$$$$-\frac{1}{\mathrm{e}}\approx -.367879\Rightarrow n\approx .382900$$$$\mathrm{and}r\in \mathbb{R}$$$$\Rightarrow$$$$r\ln r=(n-\frac{1}{2})\pi ;n\in {\mathbb{N}}^{★}$$$$\mathrm{We}\mathrm{get}\mathrm{infinite}\mathrm{solutions}$$$$n=1r\approx 2.10729948$$$$n=2r\approx 3.64417367$$$$n=3r\approx 4.92574568$$$$...$$

Commented by Humble last updated on 16/Feb/23

$$niceapproach.onecanalsogowith$$$$LambertWfunction$$

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