I have that f(z) = u(x,y) + iv(x,y) is analytic.
Under what circumstances will g(z) = u(x,y) - iv(x,y) be analytic?
Using the Cauchy-Riemann Equations,
du/dx = dv/dy and du/dy = -dv/dx
and for g to be analytic,
du/dx = -dv/dy and du/dy = dv/dx
but these equations put together imply,
dv/dy = -dv/dy, du/dy = -du/dy, etc.
It seems to me that u(x,y) and v(x,y) must be constants.
Is this true?
Note all derivatives are partial derivatives obviously.