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Bloch's Theorem (complex Variables)
In complex analysis, a branch of mathematics, Bloch's theorem describes the behaviour of holomorphic functions defined on the unit disk. It gives a lower bound on the size of a disk in which an inverse to a holomorphic function exists. It is named after André Bloch. Statement Let ''f'' be a holomorphic function in the unit disk , ''z'', ≤ 1 for which :, f'(0), =1 Bloch's Theorem states that there is a disk S ⊂ D on which f is biholomorphic and f(S) contains a disk with radius 1/72. Landau's theorem If ''f'' is a holomorphic function in the unit disk with the property , ''f′''(0), = 1, then let ''Lf'' be the radius of the largest disk contained in the image of ''f''. Landau's theorem states that there is a constant ''L'' defined as the infimum of ''Lf'' over all such functions ''f'', and that ''L'' ≥ ''B''. This theorem is named after Edmund Landau. Valiron's theorem Bloch's theorem was inspired by the following theorem of Georges Valiron: Theorem. If ''f'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), 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, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear engineering, nuclear, aerospace engineering, aerospace, mechanical engineering, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to its Taylor series (that is, it is Analyticity of holomorphic functions, analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions). History Complex analysis is one of the classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures). A bijection from the set ''X'' to the set ''Y'' has an inverse function from ''Y'' to ''X''. If ''X'' and ''Y'' are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite sets. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal D'Analyse Mathématique
The ''Journal d'Analyse Mathématique'' is a triannual peer-reviewed scientific journal published by Magnes Press (Hebrew University of Jerusalem). It was established in 1951 by Binyamin Amirà. It covers research in mathematics, especially classical analysis and related areas such as complex function theory, ergodic theory, functional analysis, harmonic analysis, partial differential equations, and quasiconformal mapping. Abstracting and indexing The journal is abstracted and indexed in: *MathSciNet *Science Citation Index Expanded *Scopus *ZbMATH Open According to the ''Journal Citation Reports'', the journal has a 2021 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 1.132. References External links *{{Official website, 1=https://www.springer.com/mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Zeitschrift .
''Mathematische Zeitschrift'' (German for ''Mathematical Journal'') is a mathematical journal for pure and applied mathematics published by Springer Verlag. It was founded in 1918 and edited by Leon Lichtenstein together with Konrad Knopp, Erhard Schmidt, and Issai Schur. Past editors include Erich Kamke, Friedrich Karl Schmidt, Rolf Nevanlinna, Helmut Wielandt, and Olivier Debarre Olivier Debarre (born 1959) is a French mathematician who specializes in complex algebraic geometry.Debarr ... External links * * Mathematics journals Publications established in 1918 {{math ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Constant
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory, statistics, and calculus. What it means for a constant to arise "naturally", and what makes a constant "interesting", is ultimately a matter of taste, with some mathematical constants being notable more for historical reasons than for their intrinsic mathematical interest. The more popular constants have been studied throughout the ages and computed to many decimal places. All named mathematical constants are definable numbers, and usually are also computable numbers (Chaitin's constant being a significant exception). Basic mathematical constants These are constants which one is likely to encounter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the image of one element of its domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an is also called a . However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see for more details. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lars Ahlfors
Lars Valerian Ahlfors (18 April 1907 – 11 October 1996) was a Finnish mathematician, remembered for his work in the field of Riemann surfaces and his text on complex analysis. Background Ahlfors was born in Helsinki, Finland. His mother, Sievä Helander, died at his birth. His father, Axel Ahlfors, was a professor of engineering at the Helsinki University of Technology. The Ahlfors family was Swedish-speaking, so he first attended the private school Nya svenska samskolan where all classes were taught in Swedish. Ahlfors studied at University of Helsinki from 1924, graduating in 1928 having studied under Ernst Lindelöf and Rolf Nevanlinna. He assisted Nevanlinna in 1929 with his work on Denjoy's conjecture on the number of asymptotic values of an entire function. In 1929 Ahlfors published the first proof of this conjecture, now known as the Denjoy–Carleman–Ahlfors theorem. It states that the number of asymptotic values approached by an entire function of order ρ alon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer , \Gamma(n) = (n-1)!\,. Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: \Gamma(z) = \int_0^\infty t^ e^\,dt, \ \qquad \Re(z) > 0\,. The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles. The gamma function has no zeroes, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function: \Gamma(z) = \mathcal M \ (z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Function Theorem
In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its ''derivative is continuous and non-zero at the point''. The theorem also gives a formula for the derivative of the inverse function. In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, giving a formula for the Jacobian matrix of the inverse. There are also versions of the inverse function theorem for complex holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth. The theorem was first established by Picard and Goursat using an iterative scheme: the basic idea is to prove a fixed point theorem using the contraction mapping theorem. Statements For functions of a single variable, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rouché's Theorem
Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions and holomorphic inside some region K with closed contour \partial K, if on \partial K, then and have the same number of zeros inside K, where each zero is counted as many times as its multiplicity. This theorem assumes that the contour \partial K is simple, that is, without self-intersections. Rouché's theorem is an easy consequence of a stronger symmetric Rouché's theorem described below. Usage The theorem is usually used to simplify the problem of locating zeros, as follows. Given an analytic function, we write it as the sum of two parts, one of which is simpler and grows faster than (thus dominates) the other part. We can then locate the zeros by looking at only the dominating part. For example, the polynomial z^5 + 3z^3 + 7 has exactly 5 zeros in the disk , z, b > 0). By the quadratic formula it has two zeros at -a \pm \sqrt. Rouché's theorem can be used to obtain mor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor's Theorem
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order ''k'' of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial. Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, although an earlier version of the result was already mentioned in 1671 by James Gregory. Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis. It gives simple arithmetic formula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
