Specifying a holomorphic function by a sequence of values

Given a sequence $(z_n, w_n)$ of pairs of complex numbers such that $|z_n| \to \infty$ as $n \to \infty$, there exists a holomorphic function $f$ such that $f(z_n) = w_n$ for all $n$.

Proof: By the Weierstrass factorization theorem, there exists a holomorphic function $f_1$ with only simple zeros precisely at each $z_n$. By the Mittag-Leffler theorem, there exists a meromorphic function $f_2$ with only simple poles precisely at each $z_n$ and with residues $w_n /f_1'(z_n)$. If $f = f_1 \cdot f_2$, then $f$ has removable singularities at each $z_n$ with values $w_n$ since \begin{align*} \lim_{z \to z_n} f(z) &= \lim_{z \to z_n} {f_1(z) - f_1(z_n) \over z-z_n} \cdot \lim_{z \to z_n}(z-z_n) f_2(z) \\[2ex] &= f_1'(z_n) \cdot \operatorname*{Res}_{z = z_n} f_2(z). \end{align*}

One use of this theorem is to give a very motivated definition of the gamma function, if you just want a holomorphic interpolation of the factorial function.

So does this theorem have a name? It is considered obvious or trivial? Is it well-known? Which complex analysis textbooks talk about it? Can it be used to prove other interesting things?

Solution 1:

I am not sure about a name for this, but in my copy of Ahlfors (3rd edition, 1979), there is an exercise on page 197 which goes like this:

"Suppose that $a_n \to \infty$ and that the $A_n$ are arbitrary complex numbers. Show that there exists an entire function $f(z)$ which satisfies $F(a_n) = A_n$."

I am not sure about the differences between your version and Ahlfors. He seems to require $a_n \to \infty$, whereas you do not, but at least the principle seems to have been well-known for some time.

Edit:

I have also found some related discussion in a question over at MathOverflow, but as with many things there, I don't understand the full implications (yet).