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, $(\forall n\in\mathbb Z^+):\frac{f^{(n)}(0)}{n!}=n!$.


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