[Apologies for a subjective, broad, and perhaps very naive 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), with special mention of any pole structure, conjectural or not, without getting into the details of that particular field.

Number theory of course offers a whole slew of examples that go under $L$-functions and 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?

Added: 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), and I was able to give a list of "classes" of functions with increasingly complicated "natural domains"

• polynomials and entire functions

• rational functions and meromorphic (on $\mathbb C$)

• algebraic functions like $\sqrt z$, and $\log z$ (on a Riemann surface)

• modular forms (on the upper half plane, often can't be extended)

For each I wanted to give nontrivial examples, so they are seeing a wide variety of functions before going into the general theory. I'm afraid too often in the standard complex variable course, we are speaking about a general holomorphic functions, and they only have polynomials or rational functions in mind when it comes to counterexamples or theorems. One of the books that tried to remedy that is

• Stalker, Complex Analysis: Fundamentals of the Classical Theory of Functions

which has the first chapter on special functions before the general theory. (I'm not sure how it would work with a course.)

So, I wanted to expand and/or enrich my list, perhaps with more specific examples that "open up" a whole subject (similar to the MO question Fundamental Examples), in a way that would be accessible to students——and hopefully also useful to mathematicians working in other areas. 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 too many examples of analytic functions, which makes it even more startling since an analytic function is completely determined by a sequence of numbers (be it Taylor series coefficients, or Dirichlet, or Fourier)––a point that perhaps is obvious but may seem surprising to students.