[Changed title as a plea to re-open the question.]
If one is to motivate a course in complex variables, what specific analytic (holomorphic/meromorphic) function of one variable would you cite as an example that comes up in other areas of mathematics? Particularly if one can comment on its natural domain (either a subset of $\mathbb C$ or an extension, i.e. a Riemann surface)—perhaps with special mention of any pole structure, conjectural or not—without getting into too much details of that particular field.
Number theory of course offers a whole slew of examples: $L$-functions, modular forms (not to forget elliptic functions), where they encapsulate a lot of very deep mathematics. Would be great to be more specific. Of course the Riemann zeta function would be high on everyone's list.
What about "special functions" that solve differential equations (on the complex domain)? I'd also love to see a layman's example of a Painlevé transcendent.
I seem to recall Weierstrass's nowhere-differentiable function was discovered as the boundary value of an analytic function. Anyone know what it was? What about the smooth but nowhere-analytic function?
ADDED later: Let me explain a bit why I thought of asking this here––and I do it with some reservation. I was trying to write up an explanatory note on analytic continuation, and found that one could get across the idea (if not the actual theorem) without having to develop the basic theory of complex variables (along the line of Cauchy). In particular, I managed to give a list of "classes" of functions with increasingly sophisticated natural domains.
polynomials and entire functions
rational functions and meromorphic (on $\mathbb C$)
algebraic functions like $\sqrt z$, and functions like $\log z$
modular forms, etc., on the upper half plane that can't be extended at all
For each I wanted to give some nontrivial examples, so the students would see a wide variety of functions before going into the general theory. ("To see" may be taken literally: https://en.wikipedia.org/wiki/Domain_coloring.) I'm afraid too often in the standard complex variable course, when we are speaking of holomorphic functions, the students could only think of polynomials or rational functions when it comes to counterexamples or "checking" theorems. One of the books that tried to remedy that is
- Stalker, Complex Analysis: Fundamentals of the Classical Theory of Functions
which denotes the first half on "special functions" before going into the general theory. (I'm not sure how it would work in an actual course.)
So, the purpose for this question is solicit help in expanding and/or enriching the above list with more examples, especially ones that "open up" a whole subject (similar to the MO question Fundamental Examples), in a way that would be accessible and beneficial to students. Not knowing much myself, I have the impression that many new examples have come up since the "classical" special functions.
I do agree with the comment that there are way too many examples of analytic functions useful in other areas, which makes it even more startling since an analytic function is completely determined by its restriction on a small neighborhood, or by a sequence of numbers (be it Taylor series coefficients, or Dirichlet, or Fourier)––a point that perhaps is obvious but may seem surprising in the standard introductory course. My impression is that we don't stress this point enough.