Recurring powers of iota

What is the value of $\huge i^{i^{i^{i^{\cdot^{\cdot^\cdot}}}}}$ ?

Clarification: $i=\sqrt{-1}$ .

#Algebra

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Hint: The expression be equal to $a+ib$ for some . Then, $a+ib=i^{(a+ib)}\implies a=e^(-\pi b/2)\cos(a\pi /2), b=e^(-\pi b/2)\sin(a\pi /2)\implies a^2+b^2=e^{-\pi b},\ b=a\tan (a\pi /2)$ . One needs to solve the resulting set of transcendental equations.

Samrat Mukhopadhyay - 4 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$