Given a partially computable function, is there an analytic complex function which is equal to it at every point of it's domain? Or under what condition does a partially computable function correspond to the restriction of an analytic complex function?
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1$\begingroup$ Maybe I'm being dense, but could you define a C.e. function? Or at least write out the name, so I can search for a definition? Given the tags and the question, I guess it's the only reasonable mathematical definition I found when searching for c.e. function: mathproservices.com/presentations/NewOrleans95/cefcndef.html $\endgroup$– Jan Jitse VenselaarMay 3, 2011 at 12:57
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$\begingroup$ I meant if course: "I guess it's not the only reasonable mathematical definition".. $\endgroup$– Jan Jitse VenselaarMay 3, 2011 at 13:03
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$\begingroup$ The questioner seems to be asking whether every computable function on $\mathbb{N}$ can be extended to a complex analytic function on $\mathbb{C}$. For example, computable functions can have crazy growth behavior, and perhaps this prevents such an extension. $\endgroup$– Joel David HamkinsMay 3, 2011 at 13:20
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3$\begingroup$ I don't know for sure what a computable function is, but if it is, among other things, a complex-valued function on the natural numbers, and you want to extend it to an entire function on $\mathbb{C}$, then I think you can. There is a theorem in Chapter 15 of Rudin's Real and Complex Analysis that guarantees for any subset $A$ of $\mathbb{C}$ with no limit points the existence of an entire function taking prescribed values at each element of $A$ (you can even specify as many derivatives as you want at each point). $\endgroup$– Keenan KidwellMay 3, 2011 at 13:47
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3$\begingroup$ This question should have said: given a computable partial function $\mathbb{N} \to \mathbb{N}$, is there a computable holomorphic extension of it to $\mathbb{C} \to \mathbb{C}$? That would then actually be an interesting question. $\endgroup$– Andrej BauerMay 3, 2011 at 16:13
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1 Answer
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Any complex-valued function on $\mathbb{N}$ can be extended to an entire function, so the answer is "yes." This follows from Theorem 15.13 of Rudin's Real and Complex Analysis, which states that for any open set $\Omega$ in $\mathbb{C}$ and subset $A$ of $\Omega$ without limit points, there exists a holomorphic function on $\Omega$ taking prescribed values at all the points of $A$.
answered May 3, 2011 at 14:15