Interesting value for W, h involving phi,Omega?

[Ivars]

If we use substitution

x^2=y

so that x= +- sqrt(y) we can get complex conjugate values if we use both roots separately:

W(-(+sqrt(y)*(pi/2) - I*(+sgrt(y)*ln(+sqrt(y)) = ln(+sqrt(y)) - i*pi/2 for y>1 but

W(-(+sqrt(y)*(pi/2) - I*(-sgrt(y)*ln(+sqrt(y)) = ln(+sqrt(y)) + i*pi/2 for y>1

than the change of sign and existance of 2 values -i,+i in h(i^-i) does not cause any questions-we just use both square roots of 1, however, not symmetrically...

Alternatively, we can decide to use only one value, but I^3 instead of I*(-1).

W(-(+sqrt(y)*(pi/2) - I^3*(+sgrt(y)*ln(+sqrt(y)) = ln(+sqrt(y)) + i*pi/2 for y>1

So we can get both branches of W by switching from i to - i in argument of W instead of taking negative square roots. From the point of view of the result of W, these are equivalent approaches.

By involving other root of y we will get h((I/x)^(x/I))= (1/I*x)

Ivars