Several complex variables




























Function
x ↦ f (x)

Examples by domain and codomain











































X → B, B → X, Bn
 → B
X → Z, Z → X
X → R, R → X, Rn → X
X → C, C → X, Cn → X


 Classes/properties 

Constant · Identity · Linear · Polynomial · Rational · Algebraic · Analytic · Smooth · Continuous · Measurable · Injective · Surjective · Bijective

  Constructions  

Restriction · Composition · λ · Inverse

  Generalizations  

Partial · Multivalued · Implicit

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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})}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 zi.


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




Contents






  • 1 Historical perspective


  • 2 Holomorphic functions


  • 3 An example on analytic continuation


  • 4 The Cn space


  • 5 See also


  • 6 Footnotes


  • 7 References


  • 8 External links





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 generalisation 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⟶C{displaystyle f:mathbf {C} ^{n}longrightarrow mathbf {C} }f:mathbf{C}^nlongrightarrowmathbf{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 generalisations 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)}f(z) defined on a domain U⊂Cn{displaystyle Usubset mathbb {C} ^{n}}Usubset mathbb {C} ^{n} is called holomorphic if f(z){displaystyle f(z)}f(z) satisfies one of the following two conditions.


(i) For each point a=(a1,…,an)∈U⊂Cn{displaystyle a=(a^{1},dots ,a^{n})in Usubset mathbb {C} ^{n}}a=(a^1,dots,a^n)in Usubsetmathbb{C}^n , f(z){displaystyle f(z)}f(z) is expressed as a power series expansion that is convergent on U{displaystyle U}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_{1}}cdots (z^{n}-a^{n})^{k_{n}} ,}f(z)=sum c_{k_1,dots,k_n}(z^1-a^1)^{k_1}cdots(z^n-a^n)^{k_n} ,









 

 

 

 

(1)


which was the origin of Weierstrass' analytic methods.

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






∂f∂z¯λ=0{displaystyle {frac {partial f}{partial {bar {z}}^{lambda }}}=0}frac{partial f}{partialbar{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^{1}dots y^{n})+iv(x^{1},dots x^{n},y^{1},dots y^{n}),}{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}),}

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









∂u∂xλ=∂v∂yλ,    ∂u∂yλ=−∂v∂xλ{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 }}}}frac{partial u}{partial x^lambda}=frac{partial v}{partial y^lambda}, frac{partial u}{partial y^lambda}=-frac{partial v}{partial x^lambda}.









 

 

 

 

(3)


Let


dzλ=dxλ+idyλ,dz¯λ=dxλ−idyλ∂∂zλ=12(∂∂xλ−i∂∂yλ),∂∂z¯λ=12(∂∂xλ+i∂∂yλ){displaystyle {begin{aligned}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{aligned}}}{displaystyle {begin{aligned}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{aligned}}}

through


Re(∂f∂z¯λ)=∂u∂xλ−∂v∂yλ=0,    Im(∂f∂z¯λ)=∂u∂yλ+∂v∂xλ=0{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}{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}

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∫|a1−w1|=r1⋯∫|an−wn|=rnf(w1,…,wn)dw1⋯dwn(w1−z1)⋯(wn−zn){displaystyle f(z^{1},dots ,z^{n})={biggl (}{frac {1}{2pi i}}{biggr )}^{n}int _{|a^{1}-w^{1}|=r_{1}}cdots int _{|a^{n}-w^{n}|=r_{n}}{frac {f(w^{1},dots ,w^{n})dw^{1}cdots dw^{n}}{(w^{1}-z^{1})cdots (w^{n}-z^{n})}}}f(z^{1},dots ,z^{n})={biggl (}{frac {1}{2pi i}}{biggr )}^{n}int _{{|a^{1}-w^{1}|=r_{1}}}cdots int _{{|a^{n}-w^{n}|=r_{n}}}{frac {f(w^{1},dots ,w^{n})dw^{1}cdots dw^{n}}{(w^{1}-z^{1})cdots (w^{n}-z^{n})}}









 

 

 

 

(4)


and then estimates the coefficients of the power series expansion ck1,…,kn{displaystyle c_{k_{1},dots ,k_{n}}}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 ri{displaystyle r_{i}}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⊂C{displaystyle G_{1},G_{2}subset mathbb {C} }G_1, G_2subsetmathbb{C} be some domains, G1∩G2{displaystyle G_{1}cap G_{2}}G_1cap G_2 connected, f1{displaystyle f_{1}}f_{1} and f2{displaystyle f_{2}}f_{2} holomorphic functions on G1,G2{displaystyle G_{1},G_{2}}G_{1},G_{2} respectively, and z0=x0+iy0∈G1∩G2{displaystyle z^{0}=x^{0}+iy^{0}in G_{1}cap G_{2}}z^0=x^0+iy^0 in G_1cap G_2 .

If f1=f2{displaystyle f_{1}=f_{2}}f_1=f_2 on {z||zj−zj0|<rj,y=y0,1≤j≤n}{displaystyle {z{bigl |}|z_{j}-z_{j}^{0}|<r_{j},y=y^{0},1leq jleq n}}{zbigl||z_j-z_j^0|<r_j,y=y^0,1le jle n} there is then a unique holomorphic function f{displaystyle f}f on G1∪G2{displaystyle G_{1}cup G_{2}}G_1cup G_2 such that f=f1{displaystyle f=f_{1}}f=f_1 on G1{displaystyle G_{1}}G_{1} and f=f2{displaystyle f=f_{2}}f=f_2 on G2{displaystyle G_{2}}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,V{displaystyle U,V}U, V be open subsets in Cn{displaystyle mathbb {C} ^{n}}mathbb {C} ^{n}, f∈O(U){displaystyle fin {mathcal {O}}(U)}f in mathcal{O}(U) and g∈O(V){displaystyle gin {mathcal {O}}(V)}g in mathcal{O}(V). Assume that U∩V≠∅{displaystyle Ucap Vneq varnothing }{displaystyle Ucap Vneq varnothing } and W{displaystyle W}W is a connected component of U∩V{displaystyle Ucap V}U cap V. If f|W=g|W{displaystyle f|_{W}=g|_{W}}f|_W =g|_W then h{displaystyle h}h is defined as


h(z)={f(z)z∈U,g(z)z∈V.{displaystyle h(z)={begin{cases}f(z)&zin U,\g(z)&zin V.end{cases}}}h(z) = begin{cases} f(z) & zin U, \ g(z) & zin V. end{cases}

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


In one variable case, n=1{displaystyle n=1}n=1, for any open domain U⫋C{displaystyle Uvarsubsetneqq mathbb {C} }U varsubsetneqq mathbb{C} there is a holomorphic function f{displaystyle f}f on U{displaystyle U}U that cannot be analytically continued beyond U{displaystyle U}U. This is because for any a∈∂U{displaystyle ain partial U}ainpartial U, f=1z−a{displaystyle f={frac {1}{z-a}}}f=frac{1}{z-a} cannot be analytically continued beyond
a{displaystyle a}a. However, in case of several variables n≥2{displaystyle ngeq 2}ngeq 2, it can occur that there exists a strictly larger open domain U~⫌U{displaystyle {widetilde {U}}varsupsetneqq U}widetilde{U} varsupsetneqq U such that all f∈O(U){displaystyle fin {mathcal {O}}(U)}finmathcal{O}(U) can be analytically continued to f~∈U~{displaystyle {tilde {f}}in {widetilde {U}}}tilde{f} inwidetilde{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 J 2 = −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 {begin{pmatrix}u&&-v\v&&uend{pmatrix}},}begin{pmatrix}<br /> u &&-v \<br /> v && u<br /> end{pmatrix},

a 2 × 2 real matrix that has the determinant


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





  1. ^ The field of complex numbers is a 2-dimensional vector space over real numbers.




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.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background: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 .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background: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 .cs1-lock-subscription a{background: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-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
    ISBN 3-7643-7490-X



External links



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



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