There is an exercise in Function Theory of One Complex Variable by Greene & Krantz that is very similar to a Proposition in the book, but I am having trouble getting to the conclusion.
Let $f : \mathbb{C} \to \mathbb{C}$ be a polynomial. Suppose further that $\frac{\partial f}{\partial z} = 0$ and $\frac{\partial f}{\partial \bar z} = 0$ for all $z \in \mathbb{C}$ Prove that $ f \equiv$ constant.
Now, part of the proposition (1.3.2) proves that if $p$ is a polynomial, $p(z, \bar z) = \sum a_{lm}z^l \bar z^m$ with $\frac{\partial f}{\partial \bar z} = 0$, then $\frac{\partial ^{l + m}}{\partial z^l \partial \bar z^m}p$ evaluated at zero is $l!m!a_{lm}$.
Is this the constant that $f$ equals? It doesn't seem like I've used the hypothesis, or reached the conclusion!
Maybe I should use the definition of the partial... $\frac{\partial f}{\partial z} := 1/2 (\frac{\partial }{\partial z} - i \cdot \frac{\partial }{\partial z})f$
Any hints would be greatly appreciated.