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The pi function, $\Pi(z)$ – defined as $\Pi(z) = \Gamma(z+1)$, where $\Gamma(z)$ is the gamma function – extends the factorial in that

$$\Pi(n) = (n)!$$

for all positive integers $n$. In other words, a function on the non-negative integers (the factorial) has been extended to the complex numbers (to form the pi function).

Is there a term for this kind of extensions, where you go from a function $f$ whose domain $D_f$ is a discrete set, i.e. consists only of isolated points, to an analytic function $g$ whose domain is more or less continuous, such that $f(n) = g(n)\ \forall\ n \in D_f$?

If not, is there a term for the more general case where you go from a function $f$ with domain $D_f$ (not necessarily discrete) to a function $g$ whose domain $D_g$ is larger that $D_f$, and such that $f(x) = g(x)\ \forall\ x \in D_f$? Can you call it "domain extension"? I guess one such extension would be analytic continuation.

Also, is there a specific technique that can be used to derive extensions from functions with discrete domains, such as the positive integers, to analytic functions with continuous domains?

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  • $\begingroup$ Perhaps "interpolation" or "an interpolating function". Regarding extending an arbitrary function from a discrete domain to an analytic function on a continuous domain, unless you have something specific in mind you're not including, isn't this a standard result in complex variables? $\endgroup$ – Dave L. Renfro Sep 28 '18 at 11:25
  • $\begingroup$ FYI, for examples of "a standard result in complex variables" that I was thinking of, see Entire function with prescribed values AND Existence of an entire function with certain property. $\endgroup$ – Dave L. Renfro Sep 28 '18 at 11:42

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