Logo
Unionpedia
Communication
Get it on Google Play
New! Download Unionpedia on your Android™ device!
Free
Faster access than browser!
 

Generalized continued fraction

Index 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 real or complex values. [1]

85 relations: Absolute convergence, Absolute value, Alfred Pringsheim, Apparent magnitude, Automorphism, Śleszyński–Pringsheim theorem, Bessel function, Bijection, Brahmagupta, Calculus, Carl Friedrich Gauss, Cauchy sequence, Common logarithm, Complex analysis, Complex-valued function, Conformal map, Continuant (mathematics), Continued fraction, Continuous function, Convergence problem, Diophantine equation, Discriminant, Doubling the cube, Equal temperament, Euclid, Euclidean algorithm, Euler's continued fraction formula, Felix Klein, Fibonacci number, Fixed point (mathematics), Function composition, Gauss's continued fraction, George Szekeres, Georges Poitou, Golden ratio, Gottfried Wilhelm Leibniz, Greatest common divisor, Helge von Koch, Hermite's problem, Hypergeometric function, Infinite compositions of analytic functions, Infinite product, Isaac Newton, Johann Heinrich Lambert, John Wallis, Joseph-Louis Lagrange, Klein polyhedron, Lattice (group), Laurent series, Leibniz formula for π, ..., Leonhard Euler, Logarithmic form, Machin-like formula, Mathematical coincidence, Mathematical induction, Möbius transformation, Meromorphic function, Monomial, Natural logarithm of 2, Neighbourhood (mathematics), Nilakantha Somayaji, Nth root, Number theory, Oskar Perron, Padé table, Palindrome, Pell's equation, Perfect fifth, Periodic continued fraction, Pi, Pietro Cataldi, Quadratic equation, Rafael Bombelli, Rate of convergence, Rational function, Residue theorem, Riemann sphere, Series (mathematics), Silver ratio, Solving quadratic equations with continued fractions, Square root, Subgroup, Twelfth root of two, Uniform convergence, Zeros and poles. Expand index (35 more) »

Absolute convergence

In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite.

New!!: Generalized continued fraction and Absolute convergence · See more »

Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

New!!: Generalized continued fraction and Absolute value · See more »

Alfred Pringsheim

Alfred Pringsheim (2 September 1850 – 25 June 1941) was a German mathematician and patron of the arts.

New!!: Generalized continued fraction and Alfred Pringsheim · See more »

Apparent magnitude

The apparent magnitude of a celestial object is a number that is a measure of its brightness as seen by an observer on Earth.

New!!: Generalized continued fraction and Apparent magnitude · See more »

Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

New!!: Generalized continued fraction and Automorphism · See more »

Śleszyński–Pringsheim theorem

In mathematics, the Śleszyński–Pringsheim theorem is a statement about convergence of certain continued fractions.

New!!: Generalized continued fraction and Śleszyński–Pringsheim theorem · See more »

Bessel function

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are the canonical solutions of Bessel's differential equation for an arbitrary complex number, the order of the Bessel function.

New!!: Generalized continued fraction and Bessel function · See more »

Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence 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.

New!!: Generalized continued fraction and Bijection · See more »

Brahmagupta

Brahmagupta (born, died) was an Indian mathematician and astronomer.

New!!: Generalized continued fraction and Brahmagupta · See more »

Calculus

Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

New!!: Generalized continued fraction and Calculus · See more »

Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

New!!: Generalized continued fraction and Carl Friedrich Gauss · See more »

Cauchy sequence

In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses.

New!!: Generalized continued fraction and Cauchy sequence · See more »

Common logarithm

In mathematics, the common logarithm is the logarithm with base 10.

New!!: Generalized continued fraction and Common logarithm · See more »

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.

New!!: Generalized continued fraction and Complex analysis · See more »

Complex-valued function

In mathematics, a complex-valued function (not to be confused with complex variable function) is a function whose values are complex numbers.

New!!: Generalized continued fraction and Complex-valued function · See more »

Conformal map

In mathematics, a conformal map is a function that preserves angles locally.

New!!: Generalized continued fraction and Conformal map · See more »

Continuant (mathematics)

In algebra, the continuant is a multivariate polynomial representing the determinant of a tridiagonal matrix and having applications in generalized continued fractions.

New!!: Generalized continued fraction and Continuant (mathematics) · See more »

Continued fraction

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.

New!!: Generalized continued fraction and Continued fraction · See more »

Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

New!!: Generalized continued fraction and Continuous function · See more »

Convergence problem

In the analytic theory of continued fractions, the convergence problem is the determination of conditions on the partial numerators ai and partial denominators bi that are sufficient to guarantee the convergence of the continued fraction x.

New!!: Generalized continued fraction and Convergence problem · See more »

Diophantine equation

In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is a solution such that all the unknowns take integer values).

New!!: Generalized continued fraction and Diophantine equation · See more »

Discriminant

In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them.

New!!: Generalized continued fraction and Discriminant · See more »

Doubling the cube

Doubling the cube, also known as the Delian problem, is an ancient geometric problem.

New!!: Generalized continued fraction and Doubling the cube · See more »

Equal temperament

An equal temperament is a musical temperament, or a system of tuning, in which the frequency interval between every pair of adjacent notes has the same ratio.

New!!: Generalized continued fraction and Equal temperament · See more »

Euclid

Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".

New!!: Generalized continued fraction and Euclid · See more »

Euclidean algorithm

. EXAMPLES CAN BE FOUND BELOW, E.G., IN THE "Matrix method" SECTION.

New!!: Generalized continued fraction and Euclidean algorithm · See more »

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.

New!!: Generalized continued fraction and Euler's continued fraction formula · See more »

Felix Klein

Christian Felix Klein (25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory.

New!!: Generalized continued fraction and Felix Klein · See more »

Fibonacci number

In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and characterized by the fact that every number after the first two is the sum of the two preceding ones: Often, especially in modern usage, the sequence is extended by one more initial term: By definition, the first two numbers in the Fibonacci sequence are either 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two.

New!!: Generalized continued fraction and Fibonacci number · See more »

Fixed point (mathematics)

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function.

New!!: Generalized continued fraction and Fixed point (mathematics) · See more »

Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.

New!!: Generalized continued fraction and Function composition · See more »

Gauss's continued fraction

In complex analysis, Gauss's continued fraction is a particular class of continued fractions derived from hypergeometric functions.

New!!: Generalized continued fraction and Gauss's continued fraction · See more »

George Szekeres

George Szekeres AM FAA (29 May 1911 – 28 August 2005) was a Hungarian–Australian mathematician.

New!!: Generalized continued fraction and George Szekeres · See more »

Georges Poitou

Georges Poitou (11 February 1926 – 14 December 1989) was a French mathematician.

New!!: Generalized continued fraction and Georges Poitou · See more »

Golden ratio

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.

New!!: Generalized continued fraction and Golden ratio · See more »

Gottfried Wilhelm Leibniz

Gottfried Wilhelm (von) Leibniz (or; Leibnitz; – 14 November 1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy.

New!!: Generalized continued fraction and Gottfried Wilhelm Leibniz · See more »

Greatest common divisor

In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.

New!!: Generalized continued fraction and Greatest common divisor · See more »

Helge von Koch

Niels Fabian Helge von Koch (25 January 1870 – 11 March 1924) was a Swedish mathematician who gave his name to the famous fractal known as the Koch snowflake, one of the earliest fractal curves to be described.

New!!: Generalized continued fraction and Helge von Koch · See more »

Hermite's problem

Hermite's problem is an open problem in mathematics posed by Charles Hermite in 1848.

New!!: Generalized continued fraction and Hermite's problem · See more »

Hypergeometric function

In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases.

New!!: Generalized continued fraction and Hypergeometric function · See more »

Infinite compositions of analytic functions

In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions.

New!!: Generalized continued fraction and Infinite compositions of analytic functions · See more »

Infinite product

In mathematics, for a sequence of complex numbers a1, a2, a3,...

New!!: Generalized continued fraction and Infinite product · See more »

Isaac Newton

Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, astronomer, theologian, author and physicist (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution.

New!!: Generalized continued fraction and Isaac Newton · See more »

Johann Heinrich Lambert

Johann Heinrich Lambert (Jean-Henri Lambert in French; 26 August 1728 – 25 September 1777) was a Swiss polymath who made important contributions to the subjects of mathematics, physics (particularly optics), philosophy, astronomy and map projections.

New!!: Generalized continued fraction and Johann Heinrich Lambert · See more »

John Wallis

John Wallis (3 December 1616 – 8 November 1703) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus.

New!!: Generalized continued fraction and John Wallis · See more »

Joseph-Louis Lagrange

Joseph-Louis Lagrange (or;; born Giuseppe Lodovico Lagrangia, Encyclopædia Britannica or Giuseppe Ludovico De la Grange Tournier, Turin, 25 January 1736 – Paris, 10 April 1813; also reported as Giuseppe Luigi Lagrange or Lagrangia) was an Italian Enlightenment Era mathematician and astronomer.

New!!: Generalized continued fraction and Joseph-Louis Lagrange · See more »

Klein polyhedron

In the geometry of numbers, the Klein polyhedron, named after Felix Klein, is used to generalize the concept of continued fractions to higher dimensions.

New!!: Generalized continued fraction and Klein polyhedron · See more »

Lattice (group)

In geometry and group theory, a lattice in \mathbbR^n is a subgroup of the additive group \mathbb^n which is isomorphic to the additive group \mathbbZ^n, and which spans the real vector space \mathbb^n.

New!!: Generalized continued fraction and Lattice (group) · See more »

Laurent series

In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree.

New!!: Generalized continued fraction and Laurent series · See more »

Leibniz formula for π

In mathematics, the Leibniz formula for pi, named after Gottfried Leibniz, states that It is also called Madhava–Leibniz series as it is a special case of a more general series expansion for the inverse tangent function, first discovered by the Indian mathematician Madhava of Sangamagrama in the 14th century, the specific case first published by Leibniz around 1676.

New!!: Generalized continued fraction and Leibniz formula for π · See more »

Leonhard Euler

Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.

New!!: Generalized continued fraction and Leonhard Euler · See more »

Logarithmic form

In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind.

New!!: Generalized continued fraction and Logarithmic form · See more »

Machin-like formula

In mathematics, Machin-like formulae are a popular technique for computing pi to a large number of digits.

New!!: Generalized continued fraction and Machin-like formula · See more »

Mathematical coincidence

A mathematical coincidence is said to occur when two expressions show a near-equality which has no known theoretical explanation.

New!!: Generalized continued fraction and Mathematical coincidence · See more »

Mathematical induction

Mathematical induction is a mathematical proof technique.

New!!: Generalized continued fraction and Mathematical induction · See more »

Möbius transformation

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0.

New!!: Generalized continued fraction and Möbius transformation · See more »

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 discrete set of isolated points, which are poles of the function.

New!!: Generalized continued fraction and Meromorphic function · See more »

Monomial

In mathematics, a monomial is, roughly speaking, a polynomial which has only one term.

New!!: Generalized continued fraction and Monomial · See more »

Natural logarithm of 2

The decimal value of the natural logarithm of 2 is approximately as shown in the first line of the table below.

New!!: Generalized continued fraction and Natural logarithm of 2 · See more »

Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.

New!!: Generalized continued fraction and Neighbourhood (mathematics) · See more »

Nilakantha Somayaji

Kelallur Nilakantha Somayaji (also referred to as Kelallur Comatiri; 14 June 1444 – 1544) was a major mathematician and astronomer of the Kerala school of astronomy and mathematics in India.

New!!: Generalized continued fraction and Nilakantha Somayaji · See more »

Nth root

In mathematics, an nth root of a number x, where n is usually assumed to be a positive integer, is a number r which, when raised to the power n yields x: where n is the degree of the root.

New!!: Generalized continued fraction and Nth root · See more »

Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

New!!: Generalized continued fraction and Number theory · See more »

Oskar Perron

Oskar Perron (7 May 1880 – 22 February 1975) was a German mathematician.

New!!: Generalized continued fraction and Oskar Perron · See more »

Padé table

In complex analysis, a Padé table is an array, possibly of infinite extent, of the rational Padé approximants to a given complex formal power series.

New!!: Generalized continued fraction and Padé table · See more »

Palindrome

A palindrome is a word, number, or other sequence of characters which reads the same backward as forward, such as madam or racecar.

New!!: Generalized continued fraction and Palindrome · See more »

Pell's equation

Pell's equation (also called the Pell–Fermat equation) is any Diophantine equation of the form where n is a given positive nonsquare integer and integer solutions are sought for x and y. In Cartesian coordinates, the equation has the form of a hyperbola; solutions occur wherever the curve passes through a point whose x and y coordinates are both integers, such as the trivial solution with x.

New!!: Generalized continued fraction and Pell's equation · See more »

Perfect fifth

In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so.

New!!: Generalized continued fraction and Perfect fifth · See more »

Periodic continued fraction

In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form x.

New!!: Generalized continued fraction and Periodic continued fraction · See more »

Pi

The number is a mathematical constant.

New!!: Generalized continued fraction and Pi · See more »

Pietro Cataldi

Pietro Antonio Cataldi (15 April 1548, Bologna – 11 February 1626, Bologna) was an Italian mathematician.

New!!: Generalized continued fraction and Pietro Cataldi · See more »

Quadratic equation

In algebra, a quadratic equation (from the Latin quadratus for "square") is any equation having the form where represents an unknown, and,, and represent known numbers such that is not equal to.

New!!: Generalized continued fraction and Quadratic equation · See more »

Rafael Bombelli

Rafael Bombelli (baptised on 20 January 1526; died 1572) was an Italian mathematician.

New!!: Generalized continued fraction and Rafael Bombelli · See more »

Rate of convergence

In numerical analysis, the speed at which a convergent sequence approaches its limit is called the rate of convergence.

New!!: Generalized continued fraction and Rate of convergence · See more »

Rational function

In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials.

New!!: Generalized continued fraction and Rational function · See more »

Residue theorem

In complex analysis, a discipline within mathematics, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals as well.

New!!: Generalized continued fraction and Residue theorem · See more »

Riemann sphere

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity.

New!!: Generalized continued fraction and Riemann sphere · See more »

Series (mathematics)

In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.

New!!: Generalized continued fraction and Series (mathematics) · See more »

Silver ratio

In mathematics, two quantities are in the silver ratio (also silver mean or silver constant) if the ratio of the sum of the smaller and twice the larger of those quantities, to the larger quantity, is the same as the ratio of the larger one to the smaller one (see below).

New!!: Generalized continued fraction and Silver ratio · See more »

Solving quadratic equations with continued fractions

In mathematics, a quadratic equation is a polynomial equation of the second degree.

New!!: Generalized continued fraction and Solving quadratic equations with continued fractions · See more »

Square root

In mathematics, a square root of a number a is a number y such that; in other words, a number y whose square (the result of multiplying the number by itself, or) is a. For example, 4 and −4 are square roots of 16 because.

New!!: Generalized continued fraction and Square root · See more »

Subgroup

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.

New!!: Generalized continued fraction and Subgroup · See more »

Twelfth root of two

The twelfth root of two or is an algebraic irrational number.

New!!: Generalized continued fraction and Twelfth root of two · See more »

Uniform convergence

In the mathematical field of analysis, uniform convergence is a type of convergence of functions stronger than pointwise convergence.

New!!: Generalized continued fraction and Uniform convergence · See more »

Zeros and poles

In mathematics, a zero of a function is a value such that.

New!!: Generalized continued fraction and Zeros and poles · See more »

Redirects here:

Determinant Formula, Determinant formula, Fundamental recurrence formulas, General continued fraction, General continued fractions, Generalised continued fraction.

References

[1] https://en.wikipedia.org/wiki/Generalized_continued_fraction

OutgoingIncoming
Hey! We are on Facebook now! »