The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space , that is, -tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space), which the Mathematics Subject Classification has as a top-level heading.
As in complex analysis of functions of one variable the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables . Equivalently, they are locally uniform limits of polynomials; or solutions to the dimensional Cauchy–Riemann equations. For one complex variable, every domain is the domain of holomorphy of some function. For several complex variables, this is not the case; there exist domains that are not the domain of holomorphy of any function, and so is not always the domain of holomorphy, so the domain of holomorphy is one of the themes in this field. Patching the local data of meromorphic functions, i.e. the problem of creating a global meromorphic function from zeros and poles, is called the Cousin problem. Also, the interesting phenomena that occur in several complex variables are fundamentally important to the study of compact complex manifolds and complex projective varieties and has a different flavour to complex analytic geometry in or on Stein manifolds, these are much similar to study of algebraic varieties that is study of the algebraic geometry than complex analytic geometry.
Historical perspective
Many examples of such functions were familiar in nineteenth-century mathematics; abelian functions, theta functions, and some hypergeometric series, and also, as an example of an inverse problem; the Jacobi inversion problem. Naturally also same 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 field 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, , E. E. Levi, 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, Karl Stein, Wilhelm Wirtinger and Francesco Severi. Hartogs proved some basic results, such as every isolated singularity is removable, for every analytic function
whenever . Naturally the analogues of contour integrals will be harder to handle; when 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 every open connected set D in we can find a function that will nowhere continue analytically over the boundary, that cannot be said for . In fact the D of that kind are rather special in nature (especially in complex coordinate spaces and Stein manifolds, satisfying 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, on the other hand, the Grauert–Riemenschneider vanishing theorem is known as a similar result for compact complex manifolds, and the Grauert–Riemenschneider conjecture is a special case of the conjecture of Narasimhan. 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, 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 , 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.
The complex coordinate space The complex coordinate space is the Cartesian product of copies of , and when is a domain of holomorphy, can be regarded as a Stein manifold, and more generalized Stein space. is also considered to be a complex projective variety, a Kähler manifold, etc. It is also an -dimensional vector space over the complex numbers, which gives its dimension over .The field of complex numbers is a 2-dimensional vector space over real numbers. Hence, as a set and as a topological space, may be identified to the real coordinate space and its topological dimension is thus .
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 , satisfying , which defines multiplication by the imaginary unit .
Any such space, as a real space, is oriented. On the complex plane thought of as a Cartesian plane, multiplication by a complex number may be represented by the real matrix
with determinant 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 to .
Holomorphic functions Definition
A function f defined on a domain and with values in is said to be holomorphic at a point if it is complex-differentiable at this point, in the sense that there exists a complex linear map such that
The function f is said to be holomorphic if it is holomorphic at all points of its domain of definition D.
If f is holomorphic, then all the partial maps :
are holomorphic as functions of one complex variable : we say that f is holomorphic in each variable separately. Conversely, if f is holomorphic in each variable separately, then f is in fact holomorphic : this is known as Hartog's theorem, or as Osgood's lemma under the additional hypothesis that f is continuous.
Cauchy–Riemann equations
In one complex variable, a function defined on the plane is holomorphic at a point if its real part and its imaginary part satisfy the so-called Cauchy-Riemann equations at namely
In several variables, a function is holomorphic if it is holomorphic in each variable separately, and hence if the real part and the imaginary part of satisfy the Cauchy Riemann equations
Using the formalism of Wirtinger derivatives, this can be reformulated as
or even more compactly using the formalism of complex differential forms, as
Cauchy's integral formula Prove the sufficiency of two conditions (A) and (B). Let f meets the conditions of being continuous and separately homorphic on domain D. Each disk has a rectifiable curve , is piecewise smoothness, class Jordan closed curve. () Let be the domain surrounded by each . Cartesian product closure is . Also, take the closed polydisc so that it becomes . and let be the center of each disk.) Using the Cauchy's integral formula of one variable repeatedly,
Because is a rectifiable Jordanian closed curveAccording to the Jordan curve theorem, domain D is bounded closed set, that is, each domain is compact. and f is continuous, so the order of products and sums can be exchanged so the iterated integral can be calculated as a multiple integral. Therefore,
Cauchy's evaluation formula Because the order of products and sums is interchangeable, from () we get
f is class -function.
From (2), if f is holomorphic, on polydisc and , the following evaluation equation is obtained.
Therefore, Liouville's theorem hold.
Power series expansion of holomorphic functions on polydisc If function f is holomorphic, on polydisc , from the Cauchy's integral formula, we can see that it can be uniquely expanded to the next power series.
In addition, f that satisfies the following conditions is called an analytic function.
For each point , is expressed as a power series expansion that is convergent on D :
We have already explained that holomorphic functions on polydisc are analytic. Also, from the theorem derived by Weierstrass, we can see that the analytic function on polydisc (convergent power series) is holomorphic.
If a sequence of functions converges uniformly on compact subsets of a domain D, the limit function f is holomorphic in D. Also, respective partial derivatives of compactly converges on domain D to the corresponding derivatives of f.
Radius of convergence of power series It is possible to define a combination of positive real numbers such that the power series converges uniformly at and does not converge uniformly at .
In this way it is possible to have a similar, combination of radius of convergenceBut there is a point where it converges outside the circle of convergence. For example if one of the variables is 0, then some terms, represented by the product of this variable, will be 0 regardless of the values taken by the other variables. Therefore, even if you take a variable that diverges when a variable is other than 0, it may converge. for a one complex variable. This combination is generally not unique and there are an infinite number of combinations.
Laurent series expansion Let be holomorphic in the polyannulus and continuous on its closure. Then
The integral in the second term, of the right-hand side is performed so as to see the zero on the left in every plane, also this integrated series is uniformly convergent in the annulus , where and , and so it is possible to integrate term.
Bochner–Martinelli formula The Bochner–Martinelli formula is a Cauchy formula that works for general domains. It states that if f is a continuously differentiable function on the closure of a domain D in with piecewise smooth boundary then
where is a differential form called the Bochner-Martinelli kernel. In particular if f is holomorphic the second term vanishes, so
Identity theorem