Hi, Henrik!

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

bo198214 Wrote:GFR Wrote: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 is equal to which is a solution to :

as is the inverse function of .

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