Question Number 223240 by fantastic last updated on 19/Jul/25
$${x}^{{x}} ={i} \\ $$$${number}\:{of}\:{solutions}?? \\ $$
Answered by mr W last updated on 19/Jul/25
$${say}\:{x}={re}^{\theta{i}} ={e}^{\mathrm{ln}\:{r}+\theta{i}} ={r}\left(\mathrm{cos}\:\theta+{i}\:\mathrm{sin}\:\theta\right)\: \\ $$$${with}\:{r},\:\theta\:\in{R}\:{and}\:{r}>\mathrm{0},\:−\pi\leqslant\theta\leqslant\pi \\ $$$${x}^{{x}} =\left({e}^{\mathrm{ln}\:{r}+\theta{i}} \right)^{{r}\left(\mathrm{cos}\:\theta+{i}\:\mathrm{sin}\:\theta\right)} ={i} \\ $$$${e}^{{r}\left(\mathrm{ln}\:{r}+\theta{i}\right)\left(\mathrm{cos}\:\theta+\mathrm{sin}\:\theta\:{i}\right)} ={i} \\ $$$${r}\left(\mathrm{ln}\:{r}+\theta{i}\right)\left(\mathrm{cos}\:\theta+\mathrm{sin}\:\theta\:{i}\right)=\left(\mathrm{2}{k}\pi+\frac{\pi}{\mathrm{2}}\right){i} \\ $$$$\left(\mathrm{ln}\:{r}\:\mathrm{cos}\:\theta−\theta\:\mathrm{sin}\:\theta\right)+\left(\mathrm{ln}\:{r}\:\mathrm{sin}\:\theta+\theta\:\mathrm{cos}\:\theta\right){i}=\left(\mathrm{2}{k}\pi+\frac{\pi}{\mathrm{2}}\right)\frac{{i}}{{r}} \\ $$$$\Rightarrow\mathrm{ln}\:{r}\:\mathrm{cos}\:\theta−\theta\:\mathrm{sin}\:\theta=\mathrm{0} \\ $$$$\Rightarrow{r}={e}^{\theta\:\mathrm{tan}\:\theta} \:\:\:\:…\left({I}\right) \\ $$$$\Rightarrow\mathrm{ln}\:{r}\:\mathrm{sin}\:\theta+\theta\:\mathrm{cos}\:\theta=\frac{\mathrm{1}}{{r}}\left(\mathrm{2}{k}\pi+\frac{\pi}{\mathrm{2}}\right) \\ $$$$\Rightarrow\theta\left(\mathrm{tan}\:\theta\:\mathrm{sin}\:\theta+\mathrm{cos}\:\theta\right)=\frac{\mathrm{1}}{{r}}\left(\mathrm{2}{k}\pi+\frac{\pi}{\mathrm{2}}\right) \\ $$$$\Rightarrow\frac{\theta{e}^{\theta\:\mathrm{tan}\:\theta} }{\mathrm{cos}\:\theta}=\mathrm{2}{k}\pi+\frac{\pi}{\mathrm{2}}\:\:\:…\left({II}\right) \\ $$$${there}\:{are}\:{infinite}\:{solutions}: \\ $$$${k}=\mathrm{0}: \\ $$$$\theta\approx−\mathrm{2}.\mathrm{924419183667},\:{r}\approx\mathrm{0}.\mathrm{524514040555} \\ $$$$\theta\approx\mathrm{0}.\mathrm{688453227108},\:{r}\approx\mathrm{1}.\mathrm{761943269397} \\ $$$${k}=\mathrm{1}: \\ $$$$\theta\approx\mathrm{0}.\mathrm{984848720826},\:{r}\approx\mathrm{4}.\mathrm{409983637648} \\ $$$${k}=\mathrm{2}: \\ $$$$\theta\approx\mathrm{1}.\mathrm{058138711897},\:{r}\approx\mathrm{6}.\mathrm{553213447542} \\ $$$$…… \\ $$