TITLE: Does there exist a holomorphic function which takes given values on the positive integers?
QUESTION [6 upvotes]: Inspired of course by What's a natural candidate for an analytic function that interpolates the tower function?
I am minded to ask what looks to me like a more natural question: given a sequence $a_1,a_2,a_3,\ldots$ of complex numbers, is there always a holomorphic function $f$ defined on the entire complex plane, with $f(n)=a_n$ for $n=1,2,3,\ldots$? No idea what the answer is myself, but wouldn't surprise me if it were well-known and even easy.

REPLY [15 votes]: This is Exercise 6, Page 26, of Knopp's Problem Book in the Theory of Functions, Volume 2: For any sequence of complex numbers $z_n$ with no finite limit point, and for any sequence of complex numbers $w_n$, there is an entire function mapping $z_n$ to $w_n$. The proof goes like this: Use the Weierstrass Factor Theorem to construct a function $W$ with simple zeros at the $z_n$. Use the Mittag-Leffler theorem to construct a function $M$ with simple poles at the $z_n$ with residues $\frac{w_n}{W'(z_n)}$. Then the function $W\cdot M$ does the job.