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 helpful 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, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation ; every complex number can be expressed in the form , where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number , a is called the real part, and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. 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.
The exponential function is a mathematical function denoted by or . Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation, but modern definitions allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to opine that the exponential function is "the most important function in mathematics".
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space Cn. The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (analytic). Holomorphic functions are the central objects of study in complex analysis.
Rolf Herman Nevanlinna was a Finnish mathematician who made significant contributions to complex analysis.
Lars Valerian Ahlfors was a Finnish mathematician, remembered for his work in the field of Riemann surfaces and his textbook on complex analysis.
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.
In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.
In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It was posed by Ludwig Bieberbach (1916) and finally proven by Louis de Branges (1985).
In the mathematical field of complex analysis, Nevanlinna theory is part of the theory of meromorphic functions. It was devised in 1925, by Rolf Nevanlinna. Hermann Weyl called it "one of the few great mathematical events of (the twentieth) century." The theory describes the asymptotic distribution of solutions of the equation f(z) = a, as a varies. A fundamental tool is the Nevanlinna characteristic T(r, f) which measures the rate of growth of a meromorphic function.
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula
In mathematics, a variable is a symbol that represents a mathematical object. A variable may represent a number, a vector, a matrix, a function, the argument of a function, a set, or an element of a set.
In the mathematical field known as complex analysis, Jensen's formula, introduced by Johan Jensen (1899), relates the average magnitude of an analytic function on a circle with the number of its zeros inside the circle. It forms an important statement in the study of entire functions.
In mathematics, the Schwarz alternating method or alternating process is an iterative method introduced in 1869–1870 by Hermann Schwarz in the theory of conformal mapping. Given two overlapping regions in the complex plane in each of which the Dirichlet problem could be solved, Schwarz described an iterative method for solving the Dirichlet problem in their union, provided their intersection was suitably well behaved. This was one of several constructive techniques of conformal mapping developed by Schwarz as a contribution to the problem of uniformization, posed by Riemann in the 1850s and first resolved rigorously by Koebe and Poincaré in 1907. It furnished a scheme for uniformizing the union of two regions knowing how to uniformize each of them separately, provided their intersection was topologically a disk or an annulus. From 1870 onwards Carl Neumann also contributed to this theory.
In mathematics, a function defined on a region of the complex plane is said to be of bounded type if it is equal to the ratio of two analytic functions bounded in that region. But more generally, a function is of bounded type in a region if and only if is analytic on and has a harmonic majorant on where . Being the ratio of two bounded analytic functions is a sufficient condition for a function to be of bounded type, and if is simply connected the condition is also necessary.
Mathematics is a broad subject that is commonly divided in many areas that may be defined by their objects of study, by the used methods, or by both. For example, analytic number theory is a subarea of number theory devoted to the use of methods of analysis for the study of natural numbers.
In mathematics, in the field of complex analysis, a Nevanlinna function is a complex function which is an analytic function on the open upper half-plane and has non-negative imaginary part. A Nevanlinna function maps the upper half-plane to itself or to a real constant, but is not necessarily injective or surjective. Functions with this property are sometimes also known as Herglotz, Pick or R functions.
Maurice Haskell Heins was an American mathematician, specializing in complex analysis and harmonic analysis.
David Drasin is an American mathematician, specializing in function theory.
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).
