Thought of this question after learning about the Lambert W function and wanted to challenge myself. Are there any complex solutions to the equation $${x}^{2^x} = {2}^{{x}^{{x}^{2}}}$$
Tried to work it out but it just kept getting more and more convoluted and disastrous, I suspect the solution will need the Lambert W function, but I don't know how to apply it. Anyone have any ideas?
There appear to be infinitely many complex solutions. Here is a plot of $\text{Re}(x^{2^x} - 2^{x^{x^2}}) = 0$ (red) and $\text{Im}(x^{2^x} - 2^{x^{x^2}}) = 0$ (blue) for $1.1 \le \text{Re}(x) \le 2.3$, $-0.6 \le \text{Im}(x) \le 0.6$. Each intersection of a red and a blue curve corresponds to a solution. This is just a small part of a complicated picture.