In complex analysis, a subfield of mathematics, a lacunary value or gap of a complex-valued function defined on a subset of the complex plane is a complex number which is not in the image of the function.
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.
Mathematics includes the study of such topics as quantity, structure, space, and change.
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.
More specifically, given a subset X of the complex plane C and a function f : X → C, a complex number z is called a lacunary value of f if z ∉ image(f).
Note, for example, that 0 is the only lacunary value of the complex exponential function. The two Picard theorems limit the number of possible lacunary values of certain types of holomorphic functions.
In mathematics, an exponential function is a function of the form
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is, at every point of its domain, complex differentiable in a neighbourhood of the point. The existence of a complex derivative in a neighbourhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal, locally, to its own Taylor series (analytic). Holomorphic functions are the central objects of study in complex analysis.
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.
In mathematics, and more specifically in naive set theory, the domain of definition of a function is the set of "input" or argument values for which the function is defined. That is, the function provides an "output" or value for each member of the domain. Conversely, the set of values the function takes on as output is termed the image of the function, which is sometimes also referred to as the range of the function.
In mathematics, a conformal map is a function that preserves orientation and angles locally. In the most common case, the function has a domain and an image in the complex plane.
In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of Dexcept for a discrete set of isolated points, which are poles of the function. This terminology comes from the Ancient Greek meros (μέρος), meaning "part," as opposed to holos (ὅλος), meaning "whole."
In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.
In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable. The concept of function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.
The theory of functions of several complex variables is the branch of mathematics dealing with complex valued functions
In mathematics, the maximum modulus principle in complex analysis states that if f is a holomorphic function, then the modulus |f | cannot exhibit a true local maximum that is properly within the domain of f.
In mathematics, infinite-dimensional holomorphy is a branch of functional analysis. It is concerned with generalizations of the concept of holomorphic function to functions defined and taking values in complex Banach spaces, typically of infinite dimension. It is one aspect of nonlinear functional analysis.
In mathematics, with special application to complex analysis, a normal family is a pre-compact family of continuous functions with respect to the compact-open topology. Informally, this means that the functions in the family are not widely spread out, but rather stick together in a somewhat "clustered" manner. It is of general interest to understand compact sets in function spaces, since these are usually truly infinite-dimensional in nature.
In complex analysis, a complex logarithm of the non-zero complex number z, denoted by w = log z, is defined to be any complex number w for which e w = z. This construction is analogous to the real logarithm function ln, which is the inverse of the real exponential function e y, satisfying e lnx = x for positive real numbers x.
In complex analysis, the open mapping theorem states that if U is a domain of the complex plane C and f : U → C is a non-constant holomorphic function, then f is an open map.
In mathematics, particularly measure theory, the essential range of a function is intuitively the 'non-negligible' range of the function: It does not change between two functions that are equal almost everywhere. One way of thinking of the essential range of a function is the set on which the range of the function is most 'concentrated'. The essential range can be defined for measurable real or complex-valued functions on a measure space.
Georgii Nikolaevich Polozii was a Soviet mathematician who mostly worked in pure mathematics such as complex analysis, approximation theory and numerical analysis. He also worked on elasticity theory, which is used in applied math and physics. He was Corresponding Member of the Academy of Sciences of the Ukrainian SSR, Doctor of Physical and Mathematical Sciences (1953), Head of the Department of Computational Mathematics of the Kyiv Cybernetics Faculty University (1958).
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