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The theory of functions of several complex variables is the branch of mathematics dealing with complex valued functions

f(z1,z2,…,zn)displaystyle f(z_1,z_2,ldots ,z_n)

on the space Cⁿ of n-tuples of complex numbers. As in complex analysis, which is the case n = 1 but of a distinct character, these are not just any functions: they are supposed to be holomorphic or complex analytic, so that locally speaking they are power series in the variables z_i.

Equivalently, as it turns out, they are locally uniform limits of polynomials; or local solutions to the n-dimensional Cauchy–Riemann equations.

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

f:Cn⟶Cdisplaystyle f:mathbf C ^nlongrightarrow mathbf C

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.

Holomorphic functions

A function f(z)displaystyle f(z) defined on a domain U⊂Cndisplaystyle Usubset mathbb C ^n is called holomorphic if f(z)displaystyle f(z) satisfies one of the following two conditions.

(i) For each point a=(a1,…,an)∈U⊂Cndisplaystyle a=(a^1,dots ,a^n)in Usubset mathbb C ^n , f(z)displaystyle f(z) is expressed as a power series expansion that is convergent on Udisplaystyle U :

f(z)=∑ck1,…,kn(z1−a1)k1⋯(zn−an)kn ,displaystyle f(z)=sum c_k_1,dots ,k_n(z^1-a^1)^k_1cdots (z^n-a^n)^k_n ,

(1)

which was the origin of Weierstrass' analytic methods.

(ii) If f(z)displaystyle f(z) is continuous on Udisplaystyle U, and for each variable zλdisplaystyle z^lambda , f(z)displaystyle f(z) is holomorphic, namely,

∂f∂z¯λ=0displaystyle frac partial fpartial bar z^lambda =0

(2)

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

zλ=xλ+iyλ,f(z1,…zn)=u(x1,…xn,y1…yn)+iv(x1,…xn,y1,…yn),displaystyle z^lambda =x^lambda +iy^lambda ,quad f(z^1,dots z^n)=u(x^1,dots x^n,y^1dots y^n)+iv(x^1,dots x^n,y^1,dots y^n),

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

∂u∂xλ=∂v∂yλ,    ∂u∂yλ=−∂v∂xλdisplaystyle frac partial upartial x^lambda =frac partial vpartial y^lambda , frac partial upartial y^lambda =-frac partial vpartial x^lambda .

(3)

Let

dzλ=dxλ+idyλ,dz¯λ=dxλ−idyλ∂∂zλ=12(∂∂xλ−i∂∂yλ),∂∂z¯λ=12(∂∂xλ+i∂∂yλ)displaystyle beginaligneddz^lambda &=dx^lambda +idy^lambda ,&dbar z^lambda &=dx^lambda -idy^lambda \frac partial partial z^lambda &=frac 12biggl (frac partial partial x^lambda -ifrac partial partial y^lambda biggr ),&frac partial partial bar z^lambda &=frac 12biggl (frac partial partial x^lambda +ifrac partial partial y^lambda biggr )endaligned

through

Re(∂f∂z¯λ)=∂u∂xλ−∂v∂yλ=0,    Im(∂f∂z¯λ)=∂u∂yλ+∂v∂xλ=0displaystyle textRebiggl (frac partial fpartial bar z^lambda biggr )=frac partial upartial x^lambda -frac partial vpartial y^lambda =0, textImbiggl (frac partial fpartial bar z^lambda biggr )=frac partial upartial y^lambda +frac partial vpartial x^lambda =0

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

f(z1,…,zn)=(12πi)n∫|w1−z1|=r1⋯∫|wn−zn|=rnf(w1,…,wn)dw1⋯dwn(w1−z1)⋯(wn−zn)displaystyle f(z^1,dots ,z^n)=biggl (frac 12pi ibiggr )^nint _=r_1cdots int _=r_nfrac f(w^1,dots ,w^n)dw^1cdots dw^n(w^1-z^1)cdots (w^n-z^n)

(4)

and then estimates the coefficients of the power series expansion ck1,…,kndisplaystyle c_k_1,dots ,k_n 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 ridisplaystyle r_i '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 G1,G2⊂Cdisplaystyle G_1,G_2subset mathbb C be some domains, G1∩G2displaystyle G_1cap G_2 connected, f1displaystyle f_1 and f2displaystyle f_2 holomorphic functions on G1,G2displaystyle G_1,G_2 respectively, and z0=x0+iy0∈G1∩G2displaystyle z^0=x^0+iy^0in G_1cap G_2 .

If f1=f2displaystyle f_1=f_2 on displaystyle z_j-z_j^0 there is then a unique holomorphic function fdisplaystyle f on G1∪G2displaystyle G_1cup G_2 such that f=f1displaystyle f=f_1 on G1displaystyle G_1 and f=f2displaystyle f=f_2 on G2displaystyle G_2 .

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.

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 U,Vdisplaystyle U,V be open subsets in Cndisplaystyle mathbb C ^n, f∈O(U)displaystyle fin mathcal O(U) and g∈O(V)displaystyle gin mathcal O(V). Assume that U∩V≠∅displaystyle Ucap Vneq varnothing and Wdisplaystyle W is a connected component of U∩Vdisplaystyle Ucap V. If f|W=g|W_W then hdisplaystyle h is defined as

h(z)={f(z)z∈U,g(z)z∈V.displaystyle h(z)=begincasesf(z)&zin U,\g(z)&zin V.endcases

The above hdisplaystyle h is called analytic continuation of fdisplaystyle f or gdisplaystyle g. Note that hdisplaystyle h is uniquely determined by the identity theorem but may be multi-valued.

In one variable case, n=1displaystyle n=1, for any open domain U⫋Cdisplaystyle Uvarsubsetneqq mathbb C there is a holomorphic function fdisplaystyle f on Udisplaystyle U that cannot be analytically continued beyond Udisplaystyle U. This is because for any a∈∂Udisplaystyle ain partial U, f=1z−adisplaystyle f=frac 1z-a cannot be analytically continued beyond atextstyle a. However, in case of several variables n≥2displaystyle ngeq 2, it can occur that there exists a strictly larger open domain U~⫌Udisplaystyle widetilde Uvarsupsetneqq U such that all f∈O(U)displaystyle fin mathcal O(U) can be analytically continued to f~∈U~displaystyle tilde fin widetilde U. This phenomenon is called Hartogs' phenomenon.

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 dimensions, where a complex structure is specified with a linear operator J (such that J² = −I) which defines the multiplication to 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

(u−vvu),displaystyle beginpmatrixu&&-v\v&&uendpmatrix,u &&-v \
v && u
endpmatrix,"/>

a 2 × 2 real matrix that has the determinant

u2+v2=|w|2.displaystyle u^2+v^2=

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.

See also

• Coherent sheaf


• Cartan's theorems A and B


• Cousin problems


• Hartogs' lemma


• Hartogs' theorem


• Biholomorphism


• Domain of holomorphy


• Complex geometry


• Complex projective space


• Several real variables


• Harmonic maps


• Harmonic morphisms

Footnotes

References

• H. Behnke and P. Thullen, Theorie der Funktionen mehrerer komplexer Veränderlichen (1934)


• 
  Salomon Bochner and W. T. Martin Several Complex Variables (1948)


• 
  Lars Hörmander, An Introduction to Complex Analysis in Several Variables (1966) and later editions


• Steven G. Krantz, Function Theory of Several Complex Variables (1992)

• R. Michael Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Springer 1986, 1998


• Volker Scheidemann, Introduction to complex analysis in several variables, Birkhäuser, 2005, .mw-parser-output cite.citationfont-style:inherit.mw-parser-output .citation qquotes:"""""""'""'".mw-parser-output .citation .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-ws-icon abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center.mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-maintdisplay:none;color:#33aa33;margin-left:0.3em.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em
  ISBN 3-7643-7490-X
  

External links

• 
  Tasty Bits of Several Complex Variables open source book by Jiří Lebl