abs(x)^2

x^2

x = 0

d/dx(x x^*) = x^* + x (assuming a function from reals to reals)

integral x x^* dx = x^3/3 + constant

min{x x^*} = 0 at x = 0

x x^* = (x abs(x)^2)/x

x x^* = x (-x + 2 Re(x))

x x^* = x (x - 2 i Im(x))