Is there any simple set of properties that uniquely characterizes differentiation in the space of complex functions?

Question

The transformation of differentiation is a linear operator over the vector space of entire functions (call this space $\mathbb{C}^E.$) Is there any simple set of properties that uniquely determines this linear operator that uses only the field structure of the complex numbers and the vector space structure of $\mathbb{C}^E$ and how they "interact" with each other, rather than using an "absolute value" that cannot be defined in terms of the field structure alone?

Answer 1

Let $D$ be a derivation on $\mathbb C^E$ such that $Dz = 1$. Since $Df = D(f \cdot 1) = Df + f D1$ for every $f$, we must have $D1 = 0$. Moreover $Dz^2 = D(z \cdot z) = 2 z Dz = 2 z$.

Now for any entire function $f(z)$ and $a \in \mathbb C$ we have $f(z) = f(a) + f'(a) (z-a) + (z-a)^2 h(z)$ where $h(z)$ is an entire function. Then $Df(z) = f'(a) + 2 (z-a) h(z) + (z-a)^2 Dh(z)$; substitute $z=a$ and this says $Df(a) = f'(a)$. But that is true for all $a\in C$, so $Df = f'$.

Of course this would also work for analytic functions in any open subset of $\mathbb C$.