I am trying to solve the following problem from Ahlfors' Complex Analysis Chapter 5, Section 2.3: Suppose that $\{a_n\}$ is a sequence of distinct complex numbers such that $a_n\to \infty$ and let $\{c_n\}$ be a sequence of arbitrary complex numbers. Show that there exists an entire function $f(z)$ satisfying $f(a_n)=c_n$.

The hint that is in Ahlfors' book is to let $g(z)$ be a function with simple zeros at each $a_n$. Such a function exists by Weierstrass' Theorem. Then, the hint says to look for appropriate $\gamma_n$ such that the following series converges $$ \sum_1^\infty g(z)\frac{e^{\gamma_n(z-a_n)}}{z-a_n}\frac{c_n}{g'(a_n)}. $$

I have been playing around with this for a while. I know that I need to find the $\gamma_n$ so that on any compact ball, $|z|\leq R$, the values

$$ \left|g(z)\frac{e^{\gamma_n(z-a_n)}}{z-a_n}\frac{c_n}{g'(a_n)}\right| $$ are bounded by some values $M_n(R)$ so that for each $R>0$, the sum $\sum M_n(R)$ converges. If I can do this then I know that the sum will converge uniformly on compact subsets, and so I know that the sum will be an analytic function, and then it will clearly have the right properties. However, I am having difficulty figuring out what I am supposed to choose for $\gamma_n$. I tried expressing $g$ as a Taylor series around each $a_n$ and then finding some upper bound of the terms in the sum which depended only on $R$, but have had no luck so far.

Can anyone provide a hint about how to go about finding such $\gamma_n$?

  • $\begingroup$ I cannot add anything constructive, but let me just remark that I am always absolutely amazed how the theory of complex functions of one variable is so much deeper than what is even hinted at in an introductory course. (Where it maybe seems that one has gained total control over holomorphic functions...) $\endgroup$ – Piotr Pstrągowski Feb 11 '13 at 1:27

Unfortunately, Ahlfors' hint is very misleading, and there is in fact a simpler way to solve this problem, especially since at this point of the book Ahlfors has proven both Mittag-Leffler and Weierstrass Theorems.

Let $g$ be an entire function with simple zeros at $a_n$. Recall that Mittag-Leffler's Theorem not only asserts the existence of meromorphic functions with poles at $a_n$, but allows us to control the singular part of the function at each $a_n$. So let $h$ be a meromorphic function on $\mathbb{C}$ with simple poles at each $a_n$ with singuar part $c_n/(z-a_n)$. Then $f:=gh$ has the desired properties.

  • 1
    $\begingroup$ Slight correction: the residues of $h$ should be $c_n/g'(a_n)$. $\endgroup$ – Unit Apr 16 '16 at 23:09

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