Question Number 101303 by Dwaipayan Shikari last updated on 01/Jul/20 | ||
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$${i}^{{i}^{{i}.\infty} } =? \\ $$ | ||
Commented by Dwaipayan Shikari last updated on 01/Jul/20 | ||
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$${If}\:{there}\:{any}\:{possible}\:{solutions}? \\ $$ | ||
Answered by mr W last updated on 01/Jul/20 | ||
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$${say}\:{i}^{{i}^{{i}^{...} } } ={x} \\ $$$${i}^{{x}} ={x} \\ $$$${e}^{\frac{\pi{ix}}{\mathrm{2}}} ={x} \\ $$$${xe}^{−\frac{\pi{ix}}{\mathrm{2}}} =\mathrm{1} \\ $$$$−\frac{\pi{ix}}{\mathrm{2}}{e}^{−\frac{\pi{ix}}{\mathrm{2}}} =−\frac{\pi{i}}{\mathrm{2}} \\ $$$$−\frac{\pi{ix}}{\mathrm{2}}={W}\left(−\frac{\pi{i}}{\mathrm{2}}\right) \\ $$$$\Rightarrow{x}=−\frac{\mathrm{2}}{\pi{i}}{W}\left(−\frac{\pi{i}}{\mathrm{2}}\right) \\ $$$$\Rightarrow{x}=\frac{\mathrm{2}{i}}{\pi}{W}\left(−\frac{\pi{i}}{\mathrm{2}}\right) \\ $$ | ||
Commented by Dwaipayan Shikari last updated on 01/Jul/20 | ||
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$${Thanking}\:{you} \\ $$ | ||
Commented by mr W last updated on 01/Jul/20 | ||
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