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One typically studies analysis in $\mathbb{R}^n$ after studying analysis in $\mathbb{R}$. Why can't the same be said $\mathbb{C}$?

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It is customary to study analysis in C^n after studying analysis in $C$. I guess I don't understand what you are asking then. – Ittay Weiss Jan 29 '13 at 3:01
I can't understand what you are asking. – Ram Jan 29 '13 at 3:27
This is a very good question that perhaps isn't so well-worded. I believe the OP is asking why SCV generally isn't taught in the standard undergraduate syllabus. – Jesse Madnick Jan 29 '13 at 4:43
As an undergraduate myself, it seemed odd to me that everyone had heard of complex analysis and analysis on $\mathbb{R}^n$, but very few students had heard of SCV, which seemed every bit as fundamental. In fact, I don't think any of my professors referenced the subject even once. – Jesse Madnick Jan 29 '13 at 4:46

3 Answers 3

up vote 56 down vote accepted

I would say that the main reason that several complex variables is rarely seen in the undergraduate curriculum (and even not that often in the graduate curriculum unless the department has some specialists in SCV) is that you can't get very far without lots of prerequisites.

You can for example start by proving Cauchy's integral formula for a polydisc, and from that Liouville's theorem and a few other well known results from one complex variable quickly follow.

From Cauchy's integral forumla, it also follows that holomorphic functions of several variables admit power series expansions (but the domain of convergence is not usually a ball in $\mathbb{C}^n$: compare $\sum_{j,k} z^j w^k$, $\sum_{k} (z+w)^k$ and $\sum_{k} z^k w^k$ for a few examples of what might happen). From here you can go on and study logarithmically convex Reinhardt domains.

Note, however, that the mere definition of a holomorphic function in several variables is a little problematic. You want to say that a function is holomorphic if is is holomorphic in each variable separately, but to show that this is equivalent to other plausible definitions (without assuming that the function is for example locally bounded or jointly continuous) is surprisingly difficult.

You may even get as far as showing a version of Hartogs' extension theorem: If $\Omega$ is a domain in $\mathbb{C^n}$ and $K$ is a compact subset such that $\Omega \setminus K$ is connected, every holomorphic function on $\Omega\setminus K$ extends to $\Omega$. (Here $n > 1$, of course.)

I think this is about how far you can get without bringing in tools from PDE, potential theory, algebra (sheaf theory), functional analyis, differential geometry, distribution theory and probably a few more fields.

The big highlight in a first course in several complex variables is usually to solve the Levi problem, i.e. to characterize the domains of existence for holomorphic functions. (Hartogs' extension theorem shows that some domains are unnatural to study, since all holomorphic functions extend to a bigger domain.) This is usually done with Hörmanders $L^2$-solution of the $\bar\partial$-equation. (Or via sheaf theory a la Oka.) While it's not strictly necessary to have a modern PDE course as a prerequisite, it's certainly valuable. At the very least you need to know some functional analysis (and preferably some potential theory as well), including some exposure to unbounded linear operators on Hilbert spaces to be able to understand the Hörmander solution. (For the sheaf theory solution, you need a healthy background in algebra instead.)

Similarly, you need some differential geometry (at least familiarity with differential forms and tangent bundles) to understand the more complicated integral forumlas such as Bochner-Martnielli's formula and the geometric aspects of pseduoconvexity, which is central for a deeper understanding of SCV. In fact, the interplay between the complex geometry of the domain and the corresponding function theory is a reoccuring theme in SCV. Function theory in strictly pseudoconvex domains, for example, look rather different from function theory in weakly pseudoconvex domains. (Many finer points concerning weakly pseudoconvex domains are still open problems.)

Summing up, to do a real meaningful course in SCV, you really need more background than what is reasonable to expect from an undergraduate. After all SCV is really a 20th century field of mathematics! The Levi problem for example wasn't solved until the early 50's (Hörmander's solution is as late as 1965).

Mind map

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Very nice answer. – Pete L. Clark Jan 29 '13 at 20:45
Please, reference some books on SCV. – Student May 31 at 9:32

The typical progression goes ${\bf R},{\bf R}^n,{\bf C},{\bf C}^n$. Different individuals get to different places along the progression.

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Well put. (+1)$\,$ – user26872 Jan 29 '13 at 4:20
I don't feel strongly enough to downvote, but I also don't feel like this answers the question [that the OP meant to ask]. I believe the OP is asking why SCV generally isn't taught in the standard undergraduate syllabus, or why people don't make a bigger deal of it. – Jesse Madnick Jan 29 '13 at 4:48
For instance, I think practically every graduate student knows the basics of analysis on $\mathbb{R}^n$ and $\mathbb{C}$. But I think relatively few get around to learning any analysis on $\mathbb{C}^n$. Why don't most graduate programs require it alongside their standard course sequences? – Jesse Madnick Jan 29 '13 at 4:51
@Jesse, I can't speak for "most graduate programs," but I can speak for my own experience. I studied several complex my second year of grad school, and I found it very hard. I'm sure I would have found it impossibly hard as an undergrad. Again speaking from my own, limited, perspective, complex is super-important in (analytic) number theory; several complex, you can get by without it. If what's true of number theory is true of other fields as well, that would argue against requiring several complex. – Gerry Myerson Jan 29 '13 at 5:02
@GerryMyerson: Yeah, that was my sense exactly: SCV seems quite difficult and not nearly as widely applicable. I'd post an answer saying as much, but I don't really understand the reasons for either of those claims. – Jesse Madnick Jan 29 '13 at 5:15

You need to know concepts from analysis on $\mathbb R$ and $\mathbb R^2$ including power series, derivatives, line integrals, functions, analytic geometry, and basic topology of those real (metric) spaces before you talk about analytic functions from $\mathbb C$ to $\mathbb C$. That is the main reason why you see analysis on real spaces before you get to analysis on $\mathbb C$. $\mathbb R^2$ with cartesian and polar coordinates is $\mathbb C^1$. And of course you do $\mathbb C^1$ before analysis on higher dimensional $\mathbb C$-spaces.

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