The K$...
variable is a parameter. It is likely generated internally by Module[]
, and it shouldn't be left in the answer. I'm not sure why you're seeing it, but it is helpful. If we replace it by k
, we can construct the following parametrized family of solutions:
mySol // ClearAll;
mySol[k_?NumericQ] :=
First@SolveValues[{ComplexExpand[
Abs[x + y*I] == Abs[(x + y*I)^(x + y*I)],
TargetFunctions -> {Re, Im}],
0 <= x <= E^k && y == Sqrt[E^(2 k) - x^2]}, {x, y}];
Then
Table[N@mySol[k], {k, 0, 5, 1/4}] // ListLinePlot