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Convergence Problem
In the analytic theory of continued fractions, the convergence problem is the determination of conditions on the partial numerators ''a''''i'' and partial denominators ''b''''i'' that are sufficient to guarantee the convergence of the infinite continued fraction : x = b_0 + \cfrac.\, This convergence problem is inherently more difficult than the corresponding problem for infinite series. Elementary results When the elements of an infinite continued fraction consist entirely of positive real numbers, the determinant formula can easily be applied to demonstrate when the continued fraction converges. Since the denominators ''B''''n'' cannot be zero in this simple case, the problem boils down to showing that the product of successive denominators ''B''''n''''B''''n''+1 grows more quickly than the product of the partial numerators ''a''1''a''2''a''3...''a''''n''+1. The convergence problem is much more difficult when the elements of the continued fraction are complex numbers. Per ... [...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 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] |
Oskar Perron
Oskar Perron (7 May 1880 – 22 February 1975) was a German mathematician. He was a professor at the University of Heidelberg from 1914 to 1922 and at the University of Munich from 1922 to 1951. He made numerous contributions to differential equations and partial differential equations, including the Perron method to solve the Dirichlet problem for elliptic partial differential equations. He wrote an encyclopedic book on continued fractions ''Die Lehre von den Kettenbrüchen''. He introduced ''Perron's paradox'' to illustrate the danger of assuming that the solution of an optimization problem exists: :''Let N be the largest positive integer. If N > 1, then N2 > N, contradicting the definition of N. Hence N = 1''. Works * ''Über die Drehung eines starren Körpers um seinen Schwerpunkt bei Wirkung äußerer Kräfte'', Diss. München 1902 * ''Grundlagen für eine Theorie der Jacobischen Kettenbruchalgorithmus'', Habilitationsschrift Leipzig 1906 * ''Die Lehre von den Ketten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chelsea Publishing Company
The Chelsea Publishing Company was a publisher of mathematical books, based in New York City New York, often called New York City (NYC), is the most populous city in the United States, located at the southern tip of New York State on one of the world's largest natural harbors. The city comprises five boroughs, each coextensive w ..., founded in 1944 by Aaron Galuten while he was still a graduate student at Columbia. Its initial focus was to republish important European works that were unavailable in the United States because of wartime restrictions, such as Hausdorff's Mengenlehre, or because the works were out of print. This soon expanded to include translations of such works into English, as well as original works by American authors. As of 1985, the company's catalog included more than 200 titles. After Galuten's death in 1994, the company was acquired in 1997 by the AMS, which continues to publish a portion of the company's original catalog under the AMS Chelse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arg (mathematics)
In mathematics (particularly in complex analysis), the argument of a complex number , denoted , is the angle between the positive real axis and the line joining the origin and , represented as a point in the complex plane, shown as \varphi in Figure 1. By convention the positive real axis is drawn pointing rightward, the positive imaginary number, imaginary axis is drawn pointing upward, and complex numbers with positive real part are considered to have an anticlockwise argument with positive sign. When any real-valued angle is considered, the argument is a multivalued function operating on the nonzero complex numbers. The principal value of this function is single-valued, typically chosen to be the unique value of the argument that lies within the interval . In this article the multi-valued function will be denoted and its principal value will be denoted , but in some sources the capitalization of these symbols is exchanged. In some older mathematical texts, the term "a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edward Burr Van Vleck
Edward Burr Van Vleck (June 7, 1863 – June 2, 1943) was an American mathematician. Early life Van Vleck was born June 7, 1863, Middletown, Connecticut. He was the son of astronomer John Monroe Van Vleck, he graduated from Wesleyan University in 1884, attended Johns Hopkins in 1885–1887, and studied at Göttingen (Ph.D., 1893). He also received 1 July 1914 an honorary doctorate of the University of Groningen (The Netherlands). He was assistant professor and professor at Wesleyan (1895–1906), and after 1906 a professor at the University of Wisconsin–Madison, where the mathematics building is named after him. His doctoral students include H. S. Wall. In 1913 he became president of the American Mathematical Society, of whose '' Transactions'' he had been first associate editor (1902–1905) and then editor (1905–1910). He was the author of ''Theory of Divergent Series and Algebraic Continued Fractions'' (1903) and of several monographs in mathematical journals. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ivan Śleszyński
Ivan () is a Slavic male given name, connected with the variant of the Greek name (English: John) from Hebrew meaning 'God is gracious'. It is associated worldwide with Slavic countries. The earliest person known to bear the name was the Bulgarian Saint Ivan of Rila. It is very popular in Russia, Ukraine, Croatia, Serbia, Bosnia and Herzegovina, Slovenia, Bulgaria, Belarus, North Macedonia, and Montenegro and has also become more popular in Romance-speaking countries since the 20th century. Etymology Ivan is the common Slavic Latin spelling, while Cyrillic spelling is two-fold: in Bulgarian, Russian, Macedonian, Serbian and Montenegrin it is , while in Belarusian and Ukrainian it is . The Old Church Slavonic (or Old Cyrillic) spelling is . It is the Slavic relative of the Latin name , corresponding to English '' John''. This Slavic version of the name originates from New Testament Greek (''Iōánnēs'') rather than from the Latin . The Greek name is in turn ... [...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''. A series satisfying the hypothesis is called '' normally convergent''. The result is often used in combination with the uniform limit theorem. Together they say that if, in addition to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Continued Fraction Formula
In the analytic theory of continued fractions, Euler's continued fraction formula is an identity connecting a certain very general infinite series with an infinite continued fraction. First published in 1748, it was at first regarded as a simple identity connecting a finite sum with a finite continued fraction in such a way that the extension to the infinite case was immediately apparent. Today it is more fully appreciated as a useful tool in analytic attacks on the general convergence problem for infinite continued fractions with complex elements. The original formula Euler derived the formula as connecting a finite sum of products with a finite continued fraction. : \begin a_0\left(1 + a_1\left(1 + a_2\left(\cdots + a_n\right)\cdots\right)\right) & = a_0 + a_0a_1 + a_0a_1a_2 + \cdots + a_0a_1a_2\cdots a_n \\ & = \cfrac\, \end The identity is easily established by induction on ''n'', and is therefore applicable in the limit: if the expression on the left is extended to rep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Inequalities
Fundamental may refer to: * Foundation of reality * Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental" * Fundamentalism, the belief in, and usually the strict adherence to, the simple or "fundamental" ideas based on faith in a system of thought * '' Fundamentals: Ten Keys to Reality'', a 2021 popular science book by Frank Wilczek * ''The Fundamentals'', a set of books important to Christian fundamentalism * Any of a number of fundamental theorems identified in mathematics, such as: ** Fundamental theorem of algebra, a theorem regarding the factorization of polynomials ** Fundamental theorem of arithmetic, a theorem regarding prime factorization * Fundamental analysis, the process of reviewing and analyzing a company's financial statements to make better economic decisions Music * Fun-Da-Mental, a rap group * ''Fundamental'' (Bonnie Raitt album), 1998 * ''Fundamental'' (Pet Shop Boys album), 2006 * ''Fundamental'' (Puya album) or the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred Pringsheim
Alfred Pringsheim (2 September 1850 – 25 June 1941) was a German mathematician and patron of the arts. He was the father-in-law of the author and Nobel Prize winner Thomas Mann. Family and academic career Pringsheim was born in Ohlau, Province of Silesia (now Oława, Poland). He came from an extremely wealthy Silesian merchant family with Jewish roots. He was the first-born child and only son of the Upper Silesian railway entrepreneur and coal mine owner Rudolf Pringsheim (1821–1901) and his wife Paula, née Deutschmann (1827–1909). He had a younger sister, Martha. Pringsheim attended the Maria Magdalena Gymnasium (school), Gymnasium in Breslau, where he excelled in music and mathematics. Starting in 1868 he studied mathematics and physics in Berlin and at the Ruprecht Karl University in Heidelberg. In 1872 he was awarded a doctorate in mathematics, studying under Leo Königsberger. In 1875, he moved from Berlin, where his parents lived, to Munich to earn his habilitati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Continued Fraction
A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, the continued fraction is finite or infinite. Different fields of mathematics have different terminology and notation for continued fraction. In number theory the standard unqualified use of the term continued fraction refers to the special case where all numerators are 1, and is treated in the article simple continued fraction. The present article treats the case where numerators and denominators are sequences \,\ of constants or functions. From the perspective of number theory, these are called generalized continued fraction. From the perspective of complex analysis or numerical analysis, however, they are just standard, and in the present article they will simply be called "continued fraction". Formulation A continued fraction is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Otto Stolz
Otto Stolz (3 July 1842 – 23 November 1905) was an Austrian mathematician noted for his work on mathematical analysis and infinitesimals. Born in Hall in Tirol, he studied at the University of Innsbruck from 1860 and the University of Vienna from 1863, receiving his habilitation there in 1867. Two years later he studied in Berlin under Karl Weierstrass, Ernst Kummer and Leopold Kronecker, and in 1871 heard lectures in Göttingen by Alfred Clebsch and Felix Klein (with whom he would later correspond), before returning to Innsbruck permanently as a professor of mathematics. His work began with geometry (on which he wrote his thesis) but after the influence of Weierstrass it shifted to real analysis, and many small useful theorems are credited to him. For example, he proved that a continuous function ''f'' on a closed interval 'a'', ''b''with midpoint convexity, i.e., f\left(\frac2\right) \leq \frac, has left and right derivatives at each point in (''a'', ''b''). He died in 1905 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
