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.

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 was able to describe 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 $\log z$ (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 the general holomorphic functions, and the students could only think polynomials or rational functions when it comes to counterexamples or "checking" theorems. One of the books that tried to remedy that is

which has the first chapter on special functions before going into the general theory. (I'm not sure how it would work in an actual 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 way too many examples of analytic functions useful in other areas, which makes it even more startling since an analytic function is very rigid and 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 in the standard introductory course. How do we convey this awe to the student is a challenge.

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.

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

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.

[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.

[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), 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? 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 was able to describe 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 $\log z$ (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 the general holomorphic functions, and the students could only think 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 has the first chapter on special functions before going into the general theory. (I'm not sure how it would work in an actual 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 way too many examples of analytic functions useful in other areas, which makes it even more startling since an analytic function is very rigid and 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 in the standard introductory course. How do we convey this awe to the student is a challenge.

[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 much in the standard complex variable course, we are speaking about a general holomorphic functions, and when it comes to theorems and counterexamples, they only have rational functions in mind. One book that tried to remedy that was

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

which has a 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 (c.f. 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.

[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.