How to construct an entire function for infinitely many prescribed values?
i.e. I hope to find an entire function $f$ given $f(z_k) = w_k$ ($w_k$ might not be zero) for a sequence of $\{z_k\}$ with no finite accumulation points.
Can anyone give me some idea? Thanks!
The usual way to do this is to combine Mittag-Leffler's theorem with the Weierstrass factorization theorem. Weierstrass gives you an entire function $g(z)$ which has a simple zero at each $z_k$. Mittag-Leffler gives you a meromorphic function $h(z)$ with a simple pole at each $z_k$ where $w_k \ne 0$, and residue $w_k/g'(z_k)$ there. After removing the removable singularities at each $z_k$, $f(z) = g(z) h(z)$ is the desired function.