|
Lacunary Series
In analysis, a lacunary function, also known as a lacunary series, is an analytic function that cannot be analytically continued anywhere outside the radius of convergence within which it is defined by a power series. The word ''lacunary'' is derived from lacuna (''pl.'' lacunae), meaning gap, or vacancy. The first known examples of lacunary functions involved Taylor series with large gaps, or lacunae, between the non-zero coefficients of their expansions. More recent investigations have also focused attention on Fourier series with similar gaps between non-zero coefficients. There is a slight ambiguity in the modern usage of the term lacunary series, which may refer to either Taylor series or Fourier series. A simple example Let a\in\mathbb\cap\left ''z'', 0 is an arbitrary positive constant, then ''f''(''z'') is a lacunary function that cannot be continued outside its circle of convergence. In other words, the sequence doesn't have to grow as fast as 2''k'' for ''f''(''z'') ... [...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] |
Antoni Zygmund
Antoni Zygmund (December 25, 1900 – May 30, 1992) was a Polish mathematician. He worked mostly in the area of mathematical analysis, including especially harmonic analysis, and he is considered one of the greatest analysts of the 20th century. Zygmund was responsible for creating the Chicago school of mathematical analysis together with his doctoral student Alberto Calderón, for which he was awarded the National Medal of Science in 1986. Biography Born in Warsaw, Zygmund obtained his Ph.D. from the University of Warsaw (1923) and was a professor at Stefan Batory University at Wilno from 1930 to 1939, when World War II broke out and Poland was occupied. In 1940 he managed to emigrate to the United States, where he became a professor at Mount Holyoke College in South Hadley, Massachusetts. In 1945–1947 he was a professor at the University of Pennsylvania, and from 1947, until his retirement, at the University of Chicago. He was a member of several scientific societies. Fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fabry Gap Theorem
In mathematics, the Fabry gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a certain "gap" between them. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic function anywhere on the boundary of its disc of convergence. The theorem may be deduced from the first main theorem of Turán's method. Statement of the theorem Let 0 < ''p''1 < ''p''2 < ... be a sequence of integers such that the sequence ''p''''n''/''n'' diverges to ∞. Let (''α''''j'')''j''∈N be a sequence of complex numbers such that the power series :f(z) = \sum_ \alpha_ z^ has radius of convergence 1. Then the unit circle is a natural boundary for the series ''f''. Converse A converse to the theorem was established by George Pólya. If lim inf ''p''''n''/''n'' is finite then there exists a power series with exponent sequence ''p''''n'', radius of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mandelbrot Set
The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. This set was first defined and drawn by Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleinian groups. Afterwards, in 1980, Benoit Mandelbrot obtained high-quality visualizations of the set while working at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York. Images of the Mandelbrot set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications; mathematically, one would say that the boundary of the Mandelbrot set is a ''fractal curve''. The "style" of this recursive detail depends on the region of the set boundary being examined. Mandelbrot set images may be created by sampling the complex numbers and testing, for each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Benoit Mandelbrot
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life". He referred to himself as a "fractalist" and is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity" in nature. In 1936, at the age of 11, Mandelbrot and his family emigrated from Warsaw, Poland, to France. After World War II ended, Mandelbrot studied mathematics, graduating from universities in Paris and in the United States and receiving a master's degree in aeronautics from the California Institute of Technology. He spent most of his career in both the United States and France, having dual French and American citizenship. In 1958, he began a 35-year career at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Szolem Mandelbrojt
Szolem Mandelbrojt (10 January 1899 – 23 September 1983) was a Polish-French mathematician who specialized in mathematical analysis. He was a professor at the Collège de France from 1938 to 1972, where he held the Chair of Analytical Mechanics and Celestial Mechanics. Biography Szolem Mandelbrojt was born on 10 January 1899 in Warsaw, Poland into a Jewish family of Lithuanian descent. He was initially educated in Warsaw, then in 1919 he moved to Kharkov, Ukraine (then USSR) and spent a year as a student of the Russian mathematician Sergei Bernstein. A year later, he emigrated to France and settled in Paris. In subsequent years, he attended the seminars of Jacques Hadamard, Henri Lebesgue, Émile Picard, and others. In 1923, he received a doctorate from Paris-Sorbonne University on the analytic continuation of the Taylor series. Hadamard was his Ph.D. advisor. In 1924 Mandelbrojt was awarded a Rockefeller Fellowship in the United States. In May 1926 he married Gladys Manuell ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent. The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of singularities. The case of several complex variables is rather different, since singularities then need not be isolated points, and its investigation was a major reason for the development of sheaf cohomology. Initial discussion Suppose ''f'' is an analytic function defined on a non-empty open subset ''U'' of the complex plane If ''V'' is a larger open subset of containing ''U'', and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaos Theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas. Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meromorphic Function
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 ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), poles of the function. The term comes from the Greek ''meros'' ( μέρος), meaning "part". Every meromorphic function on ''D'' can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on ''D'': any pole must coincide with a zero of the denominator. Heuristic description Intuitively, a meromorphic function is a ratio of two well-behaved (holomorphic) functions. Such a function will still be well-behaved, except possibly at the points where the denominator of the fraction is zero. If the denominator has a zero at ''z'' and the numerator does not, then the value of the function will approach infinity; if both parts have a zero at ''z'', then one must compare the multiplicity of these zero ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maclaurin Series
Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin (1878–1915), Australian general * Ian MacLaurin, Baron MacLaurin of Knebworth * Richard Cockburn Maclaurin (1870–1920), US physicist and educator See also * Taylor series in mathematics, a special case of which is the ''Maclaurin series'' * Maclaurin (crater), a crater on the Moon * McLaurin (other) * MacLaren (surname) * McLaren (other) McLaren is a Formula One racing team, part of the McLaren Group. McLaren or MacLaren may also refer to: * McLaren (surname) * MacLaren (surname) * Clan MacLaren, a Scottish clan Places * McLaren Flat, South Australia * McLaren Park, New Zeal ... {{surname, Maclaurin Clan MacLaren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel's Test
In mathematics, Abel's test (also known as Abel's criterion) is a method of testing for the convergence of an infinite series. The test is named after mathematician Niels Henrik Abel. There are two slightly different versions of Abel's test – one is used with series of real numbers, and the other is used with power series in complex analysis. Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions dependent on parameters. Abel's test in real analysis Suppose the following statements are true: # \sum a_n is a convergent series, # is a monotone sequence, and # is bounded. Then \sum a_nb_n is also convergent. It is important to understand that this test is mainly pertinent and useful in the context of non absolutely convergent series \sum a_n. For absolutely convergent series, this theorem, albeit true, is almost self evident. This theorem can be proved directly using summation by parts. Abel's test in complex analysis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass M-test
In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely. It applies to series whose terms are bounded functions with real or complex values, and is analogous to the comparison test for determining the convergence of series of real or complex numbers. It is named after the German mathematician Karl Weierstrass (1815-1897). Statement Weierstrass M-test. Suppose that (''f''''n'') is a sequence of real- or complex-valued functions defined on a set ''A'', and that there is a sequence of non-negative numbers (''M''''n'') satisfying the conditions * , f_n(x), \leq M_n for all n \geq 1 and all x \in A, and * \sum_^ M_n converges. Then the series :\sum_^ f_n (x) converges absolutely and uniformly on ''A''. The result is often used in combination with the uniform limit theorem. Together they say that if, in addition to the above conditions, the set ''A'' is a topological space and the functions ''f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
