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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?

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  • $\begingroup$ Theorem 15.13 in Rudin's Real and Complex Analysis is a more general version of this (you can prescribe finitely many derivatives at each of the $z_n$). $\endgroup$ – Daniel Fischer Nov 25 '13 at 23:31
  • $\begingroup$ @DanielFischer Do you know if there is a generalization to multiple variables or anything like that? $\endgroup$ – A l'Maeaux Nov 26 '13 at 21:24
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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).

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  • $\begingroup$ Good point, mine was incorrect as stated. If $|z_n| \to \infty$ does not hold, then the sequence must have an accumulation point other than $\infty$. In this case the Weierstrass and Mittag-Leffler theorems that I used clearly won't work, since holomorphic and meromorphic functions can only have zeros and poles accumulate on the boundary of their domains. $\endgroup$ – A l'Maeaux Nov 25 '13 at 23:01
  • $\begingroup$ Yes! - I was just typing an edit to my answer to that effect - I will discard it now. $\endgroup$ – Old John Nov 25 '13 at 23:03
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This is Theorem 5.2.1 in the book "Interpolation and Approximation" by Philip J. Davis. It is called Guichard's theorem.

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