# Borel's theorem for holomorphic functions

Borel's theorem states that given a sequence of real numbers $$(a_n)_{n\in \mathbb N}$$ there exists a $$C^\infty$$ function $$f\in C^\infty(\mathbb R)$$ such that $$\frac {f^{(n)}(0)}{n!}=a_n$$.

This theorem can naturally extend to complex numbers?

I mean given a sequence of complex numbers $$(c_n)_{n\in \mathbb N}$$ there exists a holomorphic function $$f\in \mathcal H( \mathbb C)$$ such that $$(\forall n\in\mathbb N):\frac {f^{(n)}(0)}{n!}=c_n$$?

If the answer is no, may we extend this theorem to special subsets complex numbers?

The answer is negative, because, in $$\mathbb C$$, every holomorphic function is analytic. Therefore, the radius of convergence of the power series $$\sum_{n=0}^\infty\frac{f^{(n)}(0)}{n!}z^n$$ must be greater than $$0$$. So, you cannot have, for instance, $$(\forall n\in\mathbb Z^+):\frac{f^{(n)}(0)}{n!}=n!$$.