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f on the space C''n'' of ''n''-'', so that locally speaking they are Power Series in the variables ''zi''. Equivalently, as it turns out, they are locally Uniform Limits of Polynomial s; or locally Square-integrable solutions to the ''n''-dimensional Cauchy-Riemann Equations . Many examples of such functions were familiar in , Theta Function s, 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 fully-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 Point s of Riemann Surface theory. With work of (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 ''). The natural domains of definition of functions, continued to the limit, are called '' Stein Manifold s'' 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 Form s of several variables, and Partial Differential Equation s. The Deformation Theory of Complex Structure s and Complex Manifold s 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 Form s. The classical candidates are the Hilbert Modular Form s and Siegel Modular Form s. These days these are associated to Algebraic Group s (respectively the Weil Restriction from a Totally Real Number Field of GL(2), and the Symplectic Group ), for which it happens that Automorphic Representation s 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. SEE ALSO REFERENCES
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