User blog comment:Sbiis Saibian/Googology101 - Part II/@comment-10429372-20141024190431/@comment-1605058-20141025070953

@Vel: I was aware of closed (using W function) form of this expression, but I wasn't sure if, by any chance, it isn't possible that some of these values overlap, just like values of W(-1/e) on branches -1 and 0 coincide.

@Sbiis: I don't know complex analysis well, so I have few questions (if answer to any of them is too long for you to bother here, please just sketch it): How do you know that for every real part of x we can find imaginary part with |10^x|=|x|? You mention function |10^(x+iy)|=|x+iy|, but in what parameter is this function? x? y? Both?

Sorry if these are silly questions, but for me it's not even at all obvious that x=10^x should have any solutions on the complex plane (that is, I don't see why W function is defined everywhere).