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Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of distinct points in $\mathbb{C}$ where $(a_n)_{n \in \mathbb{N}} \to \infty$ as $n \to \infty$. Let $(c_n)$ be a corresponding sequence of "values" in $\mathbb{C}$. Show that there exists an entire function $f$ such that for each $n \in \mathbb{N}$, $$f(a_n) = c_n \text{ in } \mathbb{C}.$$

I have absolutely no clue how to start solving this question, anyone care to drop a hint? Thanks!

JjL7
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  • https://math.stackexchange.com/questions/299818/entire-function-with-prescribed-values –  Apr 07 '20 at 14:54
  • or maybe here: https://math.stackexchange.com/questions/1364326/existence-of-an-entire-function-with-certain-property?rq=1 –  Apr 07 '20 at 14:55
  • @Paul K Thanks so much! How did you find the relevant sources so fast? – JjL7 Apr 09 '20 at 02:22
  • In math.stackexchange.com/a/303801/465015, I don't understand how $f:=gh$ satisifes $f(a_n)=c_n$, using the edited version where the residue is $c_n / g'(a_n)$. – JjL7 Apr 09 '20 at 06:00
  • At $a_n$ we have the Laurent expansions $g(z) = g'(a_n) (z - a_n) + ...$ and $h(z) = c_n / g'(a_n) \cdot (z - a_n)^{-1} + h(a_n) + h'(a_n) (z - a_n) + ...$. Now multiply both and plug in $a_n$.

    And I found them using the search function and google.

     –  Apr 09 '20 at 06:42

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