holomorphic extension

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• Holomorphic function — A rectangular grid (top) and its image under a holomorphic function f (bottom). In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex valued function of one or more complex …   Wikipedia

• Hartogs' extension theorem — In mathematics, precisely in the theory of functions of several complex variables, Hartogs extension theorem is a statement about the singularities of holomorphic functions of several variables. Informally, it states that the support of the… …   Wikipedia

• Removable singularity — In complex analysis, a removable singularity of a holomorphic function is a point at which the function is ostensibly undefined, but, upon closer examination, the domain of the function can be enlarged to include the singularity (in such a way… …   Wikipedia

• Poincaré residue — In mathematics, the Poincaré residue is a generalization, to several complex variables and complex manifold theory, of the residue at a pole of complex function theory. It is just one of a number of such possible extensions.The theory assumes… …   Wikipedia

• Complex number — A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the square root of –1. A complex… …   Wikipedia

• Hilbert space — For the Hilbert space filling curve, see Hilbert curve. Hilbert spaces can be used to study the harmonics of vibrating strings. The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It… …   Wikipedia

• Complex analysis — Plot of the function f(x)=(x2 1)(x 2 i)2/(x2 + 2 + 2i). The hue represents the function argument, while the brightness represents the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch …   Wikipedia

• Riemann surface — For the Riemann surface of a subring of a field, see Zariski–Riemann space. Riemann surface for the function ƒ(z) = √z. The two horizontal axes represent the real and imaginary parts of z, while the vertical axis represents the real… …   Wikipedia

• Floer homology — is a mathematical tool used in the study of symplectic geometry and low dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an… …   Wikipedia

• Schwarz lemma — In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions defined on the open unit disk. Lemma statementLet D = {z : |z| < 1} be the open unit disk in the complex plane C.… …   Wikipedia

• Edge-of-the-wedge theorem — In mathematics, the edge of the wedge theorem implies that holomorphic functions on two wedges with an edge in common are analytic continuations of each other provided they both give the same continuous function on the edge. It is used in quantum …   Wikipedia