can you prove that?"if F and ~F are the analytic functions on D,then f is a constant function"?

Question

D is a domain collection for f and ~f.
~f is the complement function of f function
the defination of analytica funtcion is: http://mathworld.wolfram.com/AnalyticFunction.html

function f(z) is analytic if and only if u and v satisfy the Cauchy-Riemann equations:

{partial u}/{partial x} = {partial v}/{partial y}
{partial u}/{partial y} = -{partial v}/{partial x}

f(z)=f(x+iy) = u(x,y) + i v(x,y)
(z) is analytic if and only if u and v satisfy the Cauchy-Riemann equations:

{partial u}/{partial x} = {partial v}/{partial y}
{partial u}/{partial y} = -{partial v}/{partial x}

~f(z) is the complement of f(z)

Answer 1

I'm assuming that by "complement" you mean complex conjugate and "~f" means f*.

Add the two Cauchy-Riemann equations for f with those for f* to get:

du/dx = 0 and du/dy = 0 respectively

--> u constant

Likewise subtract the two equations for f* from those for f to get:

dv/dy = 0 and dv/dx = 0 respectively

--> v constant