# Is there a term for extensions of functions on discrete sets to continuous sets?

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?

• 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? – Dave L. Renfro Sep 28 '18 at 11:25
• 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. – Dave L. Renfro Sep 28 '18 at 11:42