How it looks (i.θ)ₐ

[marraco]

\( \\[15pt]

{e^{i.\theta}} \) is an interesting and important function, so what we get if we do the same with tetration: \( \\[15pt]

{^{i.\theta}a} \)?

I put i.x in the polynomials obtained by minimizing the error \( \\[15pt]

{\Delta^2 \,=\, (^{x}a-a^{^{x-1}a})^2} \), and drawed the imaginary vs real part:




These functions probably are periodical, but my polynomials only converge on -1≤x≤1, so, be careful about looking at values outside of that range.
For example, this chart is only valid in the range 0..4, for real and -4..3 for the imaginary axis.



This is base \( \\[15pt]

{e^{-e}} \)












This is base \( \\[15pt]

{e^{\frac{1}{e}}} \)




This is a zoom in the curl transition




base e, or something close, seems to delimit the transition towards the negative axis:




Many of these seem to be described by something like \( \\[15pt]

{^{i.x}a\,=\,c+a.cos(n.x) \,+\, i. b.sin(m.x)} \)