While reading a book around complex analysis I came across a small theorem:
If $f(z)$ is a real valued, complex variabled, analytic function then it is constant.
My "Proof"
I started thinking that if this is true then given any complex variabled, complex valued, analytic function $f$ we can see that $$\Re(f), \Im(f)$$
are both real valued, complex variabled, analytic functions thus must be constant and thus $f$ is constant.
Now I know that in reality not all complex valued, complex variabled, analytic functions are constant. But I want to know what is wrong with the "proof".
Best Answer
The statement is that if $f(z)$ is a real-valued analytic function, then it is constant.
From the Cauchy-Riemann Equations, with $f(z)=u(x,y)+iv(x,y)$ and $v(x,y)\equiv0$ we have
$$\frac{\partial u(x,y)}{\partial x}=\frac{\partial v(x,y)}{\partial y}=0$$
and
$$\frac{\partial u(x,y)}{\partial y}=-\frac{\partial v(x,y)}{\partial x}=0$$
What can one deduce about $u(x,y)$ if its first partial derivatives vanish?