Consider the following function, whose Taylor series at 0 is identically zero, yet the function is not zero:.

The function goes to zero very quickly. One property of smooth functions is that they can look very different at different scales. This entry contributed by Todd Rowland. Rowland, Todd. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.

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MathWorld Book. The selection of single-valued branches using some cuts in the domain of existence and their study by the theory of single-valued analytic functions constitute one of the principal methods of studying specific multi-valued analytic functions. The non-isomorphism is brought to light on comparing the groups of automorphisms of these domains i.

Biholomorphic mapping — the groups prove to be algebraically non-isomorphic, whereas a biholomorphic mapping of one domain onto another, if it existed, would establish an isomorphism of these groups. Owing to this, the theory of biholomorphic mappings of domains in complex space is substantially different from the theory of conformal mappings in the plane.

The fundamental facts of the theory of holomorphic functions of one variable extend to holomorphic functions of several variables, sometimes in an altered form.

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This theorem is of fundamental importance in the study of analytic varieties , which are described locally, in a neighbourhood of each one of their points, as sets of common zeros of a finite number of functions which are holomorphic at this point. By Weierstrass' preparation theorem such sets may be locally described as sets of common zeros of Weierstrass polynomials, i. This description permits extensive use of algebraic methods in the local study of analytic sets. The spatial analogue of the Cauchy integral formula can be written in a particularly simple form for polycylinder domains, i.

However, polycylinder domains are only a very special class, and in general domains such a separation of variables is not possible.

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An analytic kernel, however, is essential in a number of problems, and it is therefore desirable to construct an integral formula with such a kernel for as large a class of domains as possible. An ample supply of integral formulas, including formulas with an analytic kernel for many domains, is contained in the general Leray formula. In the theory of analytic functions of several variables other integral representations, which are valid only for certain classes of domains, are also considered.

An important class of this kind consists of the so-called Weil domains, which are a generalization of the product of plane domains. For such domains one has the Bergman—Weil representation with a kernel which also depends analytically on the parameter.

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By this theorem, holomorphic functions of several variables cannot have isolated singular points. This fact is essential in the theory of multi-dimensional residues. The solution of this problem involves considerable topological and analytic difficulties. These may often be overcome by the methods proposed by E. Martinelli and J. The Leray method is more general: it is based on the examination of special homology classes and on the calculation of certain differential forms see Residue form.

The multi-dimensional theory of residues has also found applications in theoretical physics cf. Feynman integral. The Hartogs' extension theorem reveals a fundamental difference between the spatial and the planar theories. Thus arises the problem of the characterization of the natural domains of existence of holomorphic functions — the so-called domains of holomorphy. A simple sufficient condition may be formulated with the aid of the concept of a barrier at a boundary point of the domain, i. However, convexity is not a necessary condition for holomorphy: for instance a product of plane domains is always a domain of holomorphy, and such a product need not be convex.

Nevertheless, if the notion of convexity is suitably generalized, it is possible to arrive at a necessary and sufficient condition. Holomorphic convexity is a necessary and sufficient condition for a domain of holomorphy. However, this criterion is not very effective, since holomorphic convexity is difficult to verify. Another generalization is connected with the notion of a plurisubharmonic function , which is the complex analogue of a convex function.

Pseudo-convexity is also a necessary and sufficient condition for a domain to be a domain of holomorphy. For domains of the simplest types envelopes of holomorphy can be effectively constructed, but in the general case the problem is unsolvable within the class of single-sheeted domains. In the class of covering domains, the problem of constructing envelopes of holomorphy is always solvable. This problem also has applications in theoretical physics, especially in quantum field theory.

The transition from the plane to a complex space substantially increases the variety of geometrical problems related to holomorphic functions.

In particular, such functions are naturally considered not only in domains, but also on complex manifolds — smooth manifolds of even real dimension, the neighbourhood relations of which are biholomorphic. Among these, Stein manifolds cf. Stein manifold — natural generalizations of domains of holomorphy — play a special role.

Several problems in analysis may be reduced to the problem of constructing, in a given domain, a holomorphic function with given zeros or a meromorphic function with given poles and principal parts of the Laurent series.

## Analytic function by Wikipedia

In the plane case, these problems have been solved for arbitrary domains by the theorems of Weierstrass and Mittag-Leffler and their generalizations. The spatial case is different — the solvability of the corresponding problems, the so-called Cousin problems, depends on certain topological and analytic properties of the complex manifolds considered. The key step in the solution of the Cousin problems is to construct — starting from locally-defined functions with given properties — a global function, defined on the whole manifold under consideration and having the same local properties.

Such kinds of constructions are very conveniently effected using the theory of sheaves, which arose from the algebraic-topological treatment of the concept of an analytic function, and which has found important applications in various branches of mathematics. The solution of the Cousin problems by methods of the theory of sheaves was realized by H. Cartan and J. Complex analysis is one of the most important branches of analysis, it is closely connected with quite diverse branches of mathematics and it has numerous applications in theoretical physics, mechanics and technology.

Fundamental investigations on the theory of analytic functions have been carried out by Soviet mathematicians.

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Extensive interest in the theory of functions of a complex variable emerged in the Soviet Union at the beginning of the 20th century. This was in connection with noteworthy investigations by Soviet scientists on applications of the theory of analytic functions to various problems in the mechanics of continuous media. Zhukovskii and S. Chaplygin solved very important problems in hydrodynamics and aerodynamics by using methods of the theory of analytic functions. In the works of G. Kolosov and N. Muskhelishvili these methods were applied to fundamental problems in the theory of elasticity.

In subsequent years the theory of functions of a complex variable underwent extensive development. The development of various aspects of the theory of analytic functions was determined by the fundamental research of, among others, V. Golubev, N.

Luzin, I. Privalov and V. Smirnov boundary properties , M. Lavrent'ev geometric theory, quasi-conformal mappings and their applications to gas dynamics , M. Keldysh, M. Lavrent'ev and L. Sedov applications to problems in the mechanics of continuous media , D.

## Uniformly bounded derivatives implies globally analytic

Men'shov theory of monogeneity , M. Lavrent'ev and S.

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Mergelyan approximation theory , I. Vekua theory of generalized analytic functions and their applications , A. Gel'fond theory of interpolation , N. Bogolyubov and V. Vladimirov theory of analytic functions of several variables and its application to quantum field theory. Complex analysis has been an active field of research up to the present time also in the West. Some of the history may be found in the extensive article in the Encyclopaedia Britannica entitled Analysis, complex.

The modern proof, due to L. Kohn and L. Among other things, it can be used to obtain solutions of the Levi problem are domains of holomorphy the same as pseudo-convex domains? These fundamental problems had been settled previously by methods now belonging to sheaf theory , which go back to K. Oka — , and have culminated in the sheaf-cohomology theory of H. Cartan, J. Serre, H. Grauert and others, cf.