Infinite tetration and superroot of infinitesimal

[GFR]

Hi, Henrik!

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 \( x^{1/x} \).

Please don't forget the "lower branch" of y = ssqrt(x), which can be obtained via the second (-1 level) branch of Lambert Function.
(see the attachment)

By calling "plog(x)" [Product Logarithm] the logical union of the two real branches of the Lambert Function, i.e. W(0) and (W-1), i.e.:
plog(z) = W(0,z) OR W(-1,z), we may write:

y = ssqrt(x) = ln(x)/plog(ln(x))

I would be grateful if you could kindly mention this in the revised Wikipedia page. This formula can give the two upper and lower values of y = ssqrt(x). Please see also, in red, the fronteer of the x domain where y is real [x >= 1/e]. [© of GFR and KAR ....]. The apex point has coordinates:
x = e^(-1/e) = 0.692200628..
y = 1/e = 0.367879441..

Thank you in advance.

GFR