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S. Fokas, Complex Variables: Introduction and Applications (Cambridge, 2003). Ahlfors, L., Complex Analysis (McGraw-Hill, 1953). Cartan, H., Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes. (Hermann, 1961). English translation, Elementary Theory of Analytic Functions of One or Several Complex Variables.
It is also called Osgood–Brown theorem is that for holomorphic functions of several complex variables, the singularity is a accumulation point, not an isolated point. This means that the various properties that hold for holomorphic functions of one-variable complex variables do not hold for holomorphic functions of several complex variables.
Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation , complex analysis is widely applicable to two-dimensional problems in physics .
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics that investigates functions of complex numbers.It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, and electrical engineering.
A complex-valued function of several real variables may be defined by relaxing, in the definition of the real-valued functions, the restriction of the codomain to the real numbers, and allowing complex values. If f(x 1, …, x n) is such a complex valued function, it may be decomposed as
In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators [1]), named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of several complex variables, are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives ...
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers.In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves.
Typically, u and v are respectively the real and imaginary parts of a complex-valued function f(x + iy) = f(x, y) = u(x, y) + iv(x, y) of a single complex variable z = x + iy where x and y are real variables; u and v are real differentiable functions of the real variables. Then f is complex differentiable at a complex point if and only if the ...
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