Infinite tetration and superroot of infinitesimal

[Ivars]

Barrow said (not exact quote):

if x^x=1/2 has no solutions in real and complex numbers, it may lead to creation of new numbers just to fill that gap. Has anyone already tried to define such numbers?

But it has a solution in the complex numbers?

\( \frac{1}{2}=x^x \)
\( \ln\left(\frac{1}{2}\right)=x\ln(x)=e^yy \)
\( W\left(\ln\left(\frac{1}{2}\right)\right)=y=\ln(x) \)
\( x=e^{W\left(\ln\left(\frac{1}{2}\right)\right)} \)

He actually wrote he does not know if it has. "There is no square superroot of 1/2, " . And it was Bromer in 1987 article. So it has solution in complex numbers.

x= e^W(W(-ln2/2)).