Infinite tetration and superroot of infinitesimal

[bo198214]

Actually, the beautiful formula I propose is:
ssqrt(x) = ln(x) / W(ln(x)), which, for x = 1/2, gives:

ssqrt(1/2) = ln(1/2) / W(ln(1/2)) = 0.26289282802173525.. + 0.4996694356833174.. i

That reminds me to update the wikipedia tetration article as there is no formula for the square super root yet.

And indeed your formula is also a solution:
We first see that your formula \( \frac{\ln(x)}{W(\ln(x))} \) is equal to \( \frac{1}{h(1/x)} \) which is a solution to \( y^y=x \):
\( \left(\frac{1}{h(1/x)}\right)^{\frac{1}{h(1/x)}}=\frac{1}{1/x}=x \) as \( h \) is the inverse function of \( x^{1/x} \).