1. Historical Perspective

Many examples of such functions were familiar in nineteenth-century mathematics: abelian functions, theta functions, and some hypergeometric series. Naturally also any function of one variable that depends on some complex parameter is a candidate. The theory, however, for many years didn't become a full-fledged area in mathematical analysis, since its characteristic phenomena weren't uncovered. The Weierstrass preparation theorem would now be classed as commutative algebra; it did justify the local picture, ramification, that addresses the generalization of the branch points of Riemann surface theory.

With work of Friedrich Hartogs, and of Kiyoshi Oka in the 1930s, a general theory began to emerge; others working in the area at the time were Heinrich Behnke, Peter Thullen and Karl Stein. Hartogs proved some basic results, such as every isolated singularity is removable, for any analytic function

[math]\displaystyle{ f:\mathbf{C}^n\longrightarrow\mathbf{C} }[/math]

whenever n > 1. Naturally the analogues of contour integrals will be harder to handle: when n = 2 an integral surrounding a point should be over a three-dimensional manifold (since we are in four real dimensions), while iterating contour (line) integrals over two separate complex variables should come to a double integral over a two-dimensional surface. This means that the residue calculus will have to take a very different character.

After 1945 important work in France, in the seminar of Henri Cartan, and Germany with Hans Grauert and Reinhold Remmert, quickly changed the picture of the theory. A number of issues were clarified, in particular that of analytic continuation. Here a major difference is evident from the one-variable theory: while for any open connected set D in C we can find a function that will nowhere continue analytically over the boundary, that cannot be said for n > 1. In fact the D of that kind are rather special in nature (a condition called pseudoconvexity). The natural domains of definition of functions, continued to the limit, are called Stein manifolds and their nature was to make sheaf cohomology groups vanish. In fact it was the need to put (in particular) the work of Oka on a clearer basis that led quickly to the consistent use of sheaves for the formulation of the theory (with major repercussions for algebraic geometry, in particular from Grauert's work).

From this point onwards there was a foundational theory, which could be applied to analytic geometry (a name adopted, confusingly, for the geometry of zeroes of analytic functions: this is not the analytic geometry learned at school), automorphic forms of several variables, and partial differential equations. The deformation theory of complex structures and complex manifolds was described in general terms by Kunihiko Kodaira and D. C. Spencer. The celebrated paper GAGA of Serre pinned down the crossover point from géometrie analytique to géometrie algébrique.

C. L. Siegel was heard to complain that the new theory of functions of several complex variables had few functions in it, meaning that the special function side of the theory was subordinated to sheaves. The interest for number theory, certainly, is in specific generalizations of modular forms. The classical candidates are the Hilbert modular forms and Siegel modular forms. These days these are associated to algebraic groups (respectively the Weil restriction from a totally real number field of GL(2), and the symplectic group), for which it happens that automorphic representations can be derived from analytic functions. In a sense this doesn't contradict Siegel; the modern theory has its own, different directions.

Subsequent developments included the hyperfunction theory, and the edge-of-the-wedge theorem, both of which had some inspiration from quantum field theory. There are a number of other fields, such as Banach algebra theory, that draw on several complex variables.

2. Holomorphic Functions

A function [math]\displaystyle{ f(z) }[/math] defined on a domain [math]\displaystyle{ U \subset \mathbb{C}^n }[/math] is called holomorphic if [math]\displaystyle{ f(z) }[/math] satisfies one of the following two conditions.

(i) For each point [math]\displaystyle{ a=(a^1,\dots,a^n)\in U\subset\mathbb{C}^n }[/math] , [math]\displaystyle{ f(z) }[/math] is expressed as a power series expansion that is convergent on [math]\displaystyle{ U }[/math] :

${\displaystyle f(z)=\sum c_{k_1,\dots ,k_n}(z^1-a^1{)}^{k_1}\cdots (z^n-a^n{)}^{k_n},}$

which was the origin of Weierstrass' analytic methods.

(ii) If [math]\displaystyle{ f(z) }[/math] is continuous on [math]\displaystyle{ U }[/math], and for each variable [math]\displaystyle{ z^\lambda }[/math] , [math]\displaystyle{ f(z) }[/math] is holomorphic, namely,

${\displaystyle \frac{\partial f}{\partial {\overline{z}}^{\lambda }}=0}$

which is a generalization of the Cauchy–Riemann equations (using a partial Wirtinger derivative), and has the origin of Riemann's differential equation methods. (Using Hartogs' extension theorem, continuity in (ii) is not necessary.)

For each index λ let

[math]\displaystyle{ z^\lambda=x^\lambda+iy^\lambda,\quad f(z^1,\dots z^n)=u(x^1,\dots x^n,y^1\dots y^n) +iv(x^1,\dots x^n,y^1,\dots y^n), }[/math]

and generalize the usual Cauchy–Riemann equation for one variable for each index λ, then we obtain

${\displaystyle \frac{\partial u}{\partial x^{\lambda }}=\frac{\partial v}{\partial y^{\lambda }},\frac{\partial u}{\partial y^{\lambda }}=-\frac{\partial v}{\partial x^{\lambda }}}$

Let

[math]\displaystyle{ \begin{align}dz^\lambda & =dx^\lambda+idy^\lambda,& d\bar{z}^\lambda & =dx^\lambda-idy^\lambda \\ \frac{\partial}{\partial z^\lambda} & =\frac{1}{2}\biggl(\frac{\partial}{\partial x^\lambda}-i\frac{\partial}{\partial y^\lambda}\biggr), & \frac{\partial}{\partial\bar{z}^\lambda} & =\frac{1}{2}\biggl(\frac{\partial}{\partial x^\lambda}+i\frac{\partial}{\partial y^\lambda}\biggr)\end{align} }[/math]

through

[math]\displaystyle{ \text{Re}\biggl(\frac{\partial f}{\partial\bar{z}^\lambda}\biggr)=\frac{\partial u}{\partial x^\lambda}-\frac{\partial v}{\partial y^\lambda}=0,\ \ \ \ \text{Im}\biggl(\frac{\partial f}{\partial\bar{z}^\lambda}\biggr)=\frac{\partial u}{\partial y^\lambda}+\frac{\partial v}{\partial x^\lambda}=0 }[/math]

the above equations (2) and (3) turn to be equivalent.

To show that above two conditions (i) and (ii) are equivalent, it is easy to prove (i) → (ii). To prove (ii) → (i) one uses Cauchy's integral formula on the n-multiple disc for several complex variables

${\displaystyle f(z^1,\dots ,z^n)=(\frac{1}{2\pi i}{)}^n{\int }_{|w^1-z^1|=r_1}\cdots {\int }_{|w^n-z^n|=r_n}\frac{f(w^1,\dots ,w^n)d{w}^1\cdots d{w}^n}{(w^1-z^1)\cdots (w^n-z^n)}}$

and then estimates the coefficients of the power series expansion [math]\displaystyle{ c_{k_1,\dots,k_n} }[/math] in (1). While in one variable case the Cauchy's integral formula is an integral over the circumference of a disc with some radius r, in several variables case over the surface of a multiple disc with radii [math]\displaystyle{ r_i }[/math] 's as in (4).

As same as the one variable case, the identity theorem holds due to the properties of Laurent series that hold in several variable case.

Let [math]\displaystyle{ G_1, G_2\subset\mathbb{C} }[/math] be some domains, [math]\displaystyle{ G_1\cap G_2 }[/math] connected, [math]\displaystyle{ f_1 }[/math] and [math]\displaystyle{ f_2 }[/math] holomorphic functions on [math]\displaystyle{ G_1, G_2 }[/math] respectively, and [math]\displaystyle{ z^0=x^0+iy^0 \in G_1\cap G_2 }[/math] .

If [math]\displaystyle{ f_1=f_2 }[/math] on [math]\displaystyle{ \{z\bigl||z_j-z_j^0|\lt r_j,y=y^0,1\le j\le n\} }[/math] there is then a unique holomorphic function [math]\displaystyle{ f }[/math] on [math]\displaystyle{ G_1\cup G_2 }[/math] such that [math]\displaystyle{ f=f_1 }[/math] on [math]\displaystyle{ G_1 }[/math] and [math]\displaystyle{ f=f_2 }[/math] on [math]\displaystyle{ G_2 }[/math] .

Therefore, Liouville's theorem for entire functions, and the maximal principle hold for several variables. Also, the inverse function theorem and implicit function theorem hold as in the one variable case.

3. An Example on Analytic Continuation

As described in the previous there are similar results in several variables case as one variable case. However, there are very different aspects in several variable case. For example, Riemann mapping theorem, Mittag-Leffler's theorem, Weierstrass theorem, Runge's theorem and so on can not apply to the several variables case as it is in one variable case. The following example of analytic continuation in two variables shows these differences, which was one of motivations to complex analysis in several variables.

In several variables analytic continuation is defined in the same way as in one variable case. Namely, let [math]\displaystyle{ U, V }[/math] be open subsets in [math]\displaystyle{ \mathbb{C}^n }[/math], [math]\displaystyle{ f \in \mathcal{O}(U) }[/math] and [math]\displaystyle{ g \in \mathcal{O}(V) }[/math]. Assume that [math]\displaystyle{ U \cap V \ne \varnothing }[/math] and [math]\displaystyle{ W }[/math] is a connected component of [math]\displaystyle{ U \cap V }[/math]. If [math]\displaystyle{ f|_W =g|_W }[/math] then [math]\displaystyle{ h }[/math] is defined as

[math]\displaystyle{ h(z) = \begin{cases} f(z) & z\in U, \\ g(z) & z\in V. \end{cases} }[/math]

The above [math]\displaystyle{ h }[/math] is called analytic continuation of [math]\displaystyle{ f }[/math] or [math]\displaystyle{ g }[/math]. Note that [math]\displaystyle{ h }[/math] is uniquely determined by the identity theorem but may be multi-valued.

In one variable case, [math]\displaystyle{ n=1 }[/math], for any open domain [math]\displaystyle{ U \varsubsetneqq \mathbb{C} }[/math] there is a holomorphic function [math]\displaystyle{ f }[/math] on [math]\displaystyle{ U }[/math] that cannot be analytically continued beyond [math]\displaystyle{ U }[/math]. This is because for any [math]\displaystyle{ a\in\partial U }[/math], [math]\displaystyle{ f=\frac{1}{z-a} }[/math] cannot be analytically continued beyond [math]\displaystyle{ a }[/math]. However, in case of several variables [math]\displaystyle{ n\ge 2 }[/math], it can occur that there exists a strictly larger open domain [math]\displaystyle{ \widetilde{U} \varsupsetneqq U }[/math] such that all [math]\displaystyle{ f\in\mathcal{O}(U) }[/math] can be analytically continued to [math]\displaystyle{ \tilde{f} \in\widetilde{U} }[/math]. This phenomenon is called Hartogs' phenomenon.

4. The Cⁿ Space

The simplest Stein manifold is the space Cⁿ (the complex n-space), which consists of n-tuples of complex numbers. It can be considered as an n-dimensional vector space over complex numbers, which gives its dimension 2n over R.^[1] Hence, as a set, and as topological space, Cⁿ is identical to R²ⁿ and its topological dimension is 2n.

In coordinate-free language, any vector space over complex numbers may be thought of as a real vector space of twice as many dimensions, where a complex structure is specified by a linear operator J (such that J ² = −I) which defines multiplication by the imaginary unit i.

Any such space, as a real space, is oriented. On the complex plane thought of as the Cartesian plane, multiplication to a complex number w = u + iv has the real matrix

[math]\displaystyle{ \begin{pmatrix} u & -v \\ v & u \end{pmatrix}, }[/math]

a 2 × 2 real matrix that has the determinant

[math]\displaystyle{ u^2 + v^2 = |w|^2\,. }[/math]

Likewise, if one expresses any finite-dimensional complex linear operator as a real matrix (which will be composed from 2 × 2 blocks of the aforementioned form), then its determinant equals to the square of absolute value of the corresponding complex determinant. It is a non-negative number, which implies that the (real) orientation of the space is never reversed by a complex operator. The same applies to Jacobians of holomorphic functions from Cⁿ to Cⁿ.