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Chain Sequence
In the analytic theory of continued fractions, a chain sequence is an infinite sequence of non-negative real numbers chained together with another sequence of non-negative real numbers by the equations : a_1 = (1-g_0)g_1 \quad a_2 = (1-g_1)g_2 \quad a_n = (1-g_)g_n where either (a) 0 ≤ ''g''''n'' < 1, or (b) 0 < ''g''''n'' ≤ 1. Chain sequences arise in the study of the – both in connection with the , and also as part of the theory of continued fractions. The infinite continue ... [...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, 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, aerospace, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to its Taylor series (that is, it is 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 branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Continued Fraction
In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values. A generalized continued fraction is an expression of the form :x = b_0 + \cfrac where the () are the partial numerators, the are the partial denominators, and the leading term is called the ''integer'' part of the continued fraction. The successive convergents of the continued fraction are formed by applying the fundamental recurrence formulas: :\begin x_0 &= \frac = b_0, \\ pxx_1 &= \frac = \frac, \\ pxx_2 &= \frac = \frac,\ \dots \end where is the ''numerator'' and is the ''denominator'', called continuants, of the th convergent. They are given by the recursion :\begin A_n &= b_n A_ + a_n A_, \\ B_n &= b_n B_ + a_n B_ \qquad \text n \ge 1 \end with initial values :\begin A_ &= 1,& A_0&=b_0,\\ B_&=0, & B_0&=1. \end If the sequen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 continued fraction : x = b_0 + \cfrac.\, This convergence problem for continued fractions is inherently more difficult than the corresponding convergence 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 fracti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parabola Theorem
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface. The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the "vertex" and is the point where the parabola is most sharply curved. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a fixed field , such as the real or complex numbers, and one speaks of a quadratic form over . If K=\mathbb R, and the quadratic form takes zero only when all variables are simultaneously zero, then it is a definite quadratic form, otherwise it is an isotropic quadratic form. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry ( Riemannian metric, second fundamental form), differential topology ( intersection forms of four-manifolds), and Lie theory (the Killing form). Quadratic forms are not to be confused with a quadratic equation, which has only one variable and includes terms of degree two or less. A quadra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hubert Stanley Wall
Hubert Stanley Wall (December 2, 1902 – September 12, 1971) was an American mathematician who worked primarily in the field of continued fractions. He is also known as one of the leading proponents of the Moore method of teaching. Early life and education Wall was born in Rockwell City, Iowa on December 2, 1902. He received the Bachelor of Arts and Master of Arts degrees from Cornell College in Mount Vernon, Iowa in 1924. He received his Ph.D. degree from the University of Wisconsin (now University of Wisconsin–Madison) in 1927. He married Mary Kate Parker, a lawyer and Texas assistant Attorney General. Her specialty was election law. Career Upon receiving his Ph.D. Wall joined the faculty at Northwestern University and stayed until 1944 except for the academic year 1938–1939 when he was at the Institute for Advanced Study. He then went to the Illinois Institute of Technology for two years before moving in 1946 to the University of Texas where he spent the rest of hi ... [...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 Ke ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
