Let $f$ be an entire function. Prove or disprove The real part of $f$ is infinitely many times differentiable or not.
I think the result is true.
Since $f$ is an entire function by Cauchy Riemann equations $f'(x,y)=u_x(x,y)+iv_x(x,y)=v_y(x,y)-iu_y(x,y)$.
Now since $f$ is an entire function by Cauchy Integral Formula for derivatives, $f$ is many times differentiable and applying Cauchy Riemann equations repeatedly we can obtained the partial derivatives of all orders of $u(x,y)$ and they all are continuous. So $u$ is infinitely many times differentiable. Am I correct?
Best Answer
Yes that is the right idea. A nice approach is by contradiction. If $Re(f)$ was not infinitely many times differentiable, then there is some $n$ such that $Re^{(n)}(f)$ is not differentiable, that is taking the partial by $x$ n times. But this means that the $n$th derivative of $f$ does not exist by the Cauchy Riemann equations, a contradiction since $f$ is entire hence infinitely differentiable.