Information Discussion 0 Files Holdings. Series Dover Books on Mathematics Subject category Mathematical Physics and Mathematics Abstract Noted mathematician offers basic treatment of theory of analytic functions of a complex variable, touching on analytic functions of several real or complex variables as well as the existence theorem for solutions of differential systems where data is analytic.

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Also included is a systematic, though elementary, exposition of theory of abstract complex manifolds of one complex dimension. Topics include power series in one variable, holomorphic functions, Cauchy's integral, more. Back to search.

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Record created , last modified With work of Friedrich Hartogs , and of Kiyoshi Oka in the s, 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. This means that the residue calculus will have to take a very different character.

## Analytic Functions of Several Complex Variables

After 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. 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.

## Several complex variables

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

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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. 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.

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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.