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Phragmén–Lindelöf Principle
In complex analysis, the Phragmén–Lindelöf principle (or method), first formulated by Lars Edvard Phragmén (1863–1937) and Ernst Leonard Lindelöf (1870–1946) in 1908, is a technique which employs an auxiliary, parameterized function to prove the boundedness of a holomorphic function f (i.e., , f(z), Background In the theory of complex functions, it is known that the modulus (absolute value) of a |
Complex Analysis
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, and 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. As a differentiable function of a complex variable is equal to the sum function given by its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable, that is, '' holomorphic functions''. The concept can be extended to functions of several complex variables. Complex analysis is contrasted with real analysis, which dea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hadamard Three-lines Theorem
In complex analysis, a branch of mathematics, the Hadamard three-line theorem is a result about the behaviour of holomorphic functions defined in regions bounded by parallel lines in the complex plane. The theorem is named after the French mathematician Jacques Hadamard. Statement Define F(z) by : F(z)=f(z) M(a)^M(b)^ where , F(z), \leq 1 on the edges of the strip. The result follows once it is shown that the inequality also holds in the interior of the strip. After an affine transformation in the coordinate z, it can be assumed that a = 0 and b = 1. The function : F_n(z) = F(z) e^e^ tends to 0 as , z, tends to infinity and satisfies , F_n, \leq 1 on the boundary of the strip. The maximum modulus principle can therefore be applied to F_n in the strip. So , F_n(z), \leq 1. Because F_n(z) tends to F(z) as n tends to infinity, it follows that , F(z), \leq 1. ∎ Applications The three-line theorem can be used to prove the Hadamard three-circle theorem for a bound ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Carlson's Theorem
In mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem which was discovered by Fritz David Carlson. Informally, it states that two different analytic functions which do not grow very fast at infinity can not coincide at the integers. The theorem may be obtained from the Phragmén–Lindelöf theorem, which is itself an extension of the maximum-modulus theorem. Carlson's theorem is typically invoked to defend the uniqueness of a Newton series expansion. Carlson's theorem has generalized analogues for other expansions. Statement Assume that satisfies the following three conditions. The first two conditions bound the growth of at infinity, whereas the third one states that vanishes on the non-negative integers. # is an entire function of exponential type, meaning that , f(z), \leq C e^, \quad z \in \mathbb for some real values , . # There exists such that , f(iy), \leq C e^, \quad y \in \mathbb # for every non-negative integer . Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Real Part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Riemann Sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value \infty for infinity. With the Riemann model, the point \infty is near to very large numbers, just as the point 0 is near to very small numbers. The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0=\infty well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the Pole (complex analysis), poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere. In geometr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Circular Sector
A circular sector, also known as circle sector or disk sector or simply a sector (symbol: ⌔), is the portion of a disk (a closed region bounded by a circle) enclosed by two radii and an arc, with the smaller area being known as the ''minor sector'' and the larger being the ''major sector''. In the diagram, is the central angle, the radius of the circle, and is the arc length of the minor sector. The angle formed by connecting the endpoints of the arc to any point on the circumference that is not in the sector is equal to half the central angle. Types A sector with the central angle of 180° is called a '' half-disk'' and is bounded by a diameter and a semicircle. Sectors with other central angles are sometimes given special names, such as quadrants (90°), sextants (60°), and octants (45°), which come from the sector being one quarter, sixth or eighth part of a full circle, respectively. Area The total area of a circle is . The area of the sector can be obtai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Subharmonic Function
In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory. Intuitively, subharmonic functions are related to convex functions of one variable as follows. If the graph of a convex function and a line intersect at two points, then the graph of the convex function is ''below'' the line between those points. In the same way, if the values of a subharmonic function are no larger than the values of a harmonic function on the ''boundary'' of a ball, then the values of the subharmonic function are no larger than the values of the harmonic function also ''inside'' the ball. ''Superharmonic'' functions can be defined by the same description, only replacing "no larger" with "no smaller". Alternatively, a superharmonic function is just the negative of a subharmonic function, and for this reason any property of subharmonic functions can be easily transferred to superharm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lars Edvard Phragmén
Lars Edvard Phragmén (2 September 1863, Örebro – 14 March 1937) was a Swedish mathematician who made contributions to complex analysis, voting theory, and actuarial science. He succeeded Sofia Kovalevskaia as professor of mathematical analysis at Stockholm University in 1892, where his research culminated in the development of the Phragmén–Lindelöf principle, and later served as president of the board of the Mittag-Leffler Institute. His pioneering "load-balancing" voting methods for proportional representation have experienced renewed interest in modern social choice theory and found practical application in Swedish parliamentary elections. Early life and career He was the son of a college professor and studied at Uppsala University and Stockholm University, graduating from Uppsala in 1889. He succeeded Sofia Kovalevskaia as professor of mathematical analysis at Stockholm University in 1892, and in 1884 provided a new proof of the Cantor–Bendixson theorem. His resear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Continuous Function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bounded Set
In mathematical analysis and related areas of mathematics, a set is called bounded if all of its points are within a certain distance of each other. Conversely, a set which is not bounded is called unbounded. The word "bounded" makes no sense in a general topological space without a corresponding metric. '' Boundary'' is a distinct concept; for example, a circle (not to be confused with a disk) in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. A bounded set is not necessarily a closed set and vice versa. For example, a subset of a 2-dimensional real space constrained by two parabolic curves and defined in a Cartesian coordinate system is closed by the curves but not bounded (so unbounded). Definition in the real numbers A set of real numbers is called ''bounded from above'' if there exists some real number (not necessarily in ) such that for all in . The number is called an upper bound of . The terms ''bounded from b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
