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Somos' Quadratic Recurrence Constant
In mathematics, Somos' quadratic recurrence constant, named after Michael Somos, is the number :\sigma = \sqrt = 1^\;2^\; 3^ \cdots.\, This can be easily re-written into the far more quickly converging product representation :\sigma = \sigma^2/\sigma = \left(\frac \right)^ \left(\frac \right)^ \left(\frac \right)^ \left(\frac \right)^ \cdots, which can then be compactly represented in infinite product form by: :\sigma = \prod_^ \left(1 + \frac\right)^. The constant σ arises when studying the asymptotic behaviour of the sequence :g_0 = 1\, ; \,g_n = n g_^2, \qquad n > 1,\, with first few terms 1, 1, 2, 12, 576, 1658880, ... . This sequence can be shown to have asymptotic behaviour as follows: :g_n \sim \frac . Guillera and Sondow give a representation in terms of the derivative of the Lerch transcendent: :\ln \sigma = \frac \frac\!\left( \frac, 0, 1 \right) where ln is the natural logarithm and \Phi(''z'', ''s'', ''q'') is the Lerch transcendent. Finall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Somos
Michael Somos is an American mathematician, who was a visiting scholar in the Georgetown University Mathematics and Statistics department for four years and is a visiting scholar at Catholic University of America. In the late eighties he proposed a conjecture about certain polynomial recurrences, now called Somos sequences, that surprisingly in some cases contain only integers. Somos' quadratic recurrence constant is also named after him. Notes References * Michael Somos and Robert Haas, "A Linked Pair of Sequences Implies the Primes Are Infinite", ''The American Mathematical Monthly'', volume 110, number 6 (June – July, 2003), pp. 539–540 External links Michael Somos's homepage '' |
Limit (mathematics)
In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. In formulas, a limit of a function is usually written as : \lim_ f(x) = L, (although a few authors may use "Lt" instead of "lim") and is read as "the limit of of as approaches equals ". The fact that a function approaches the limit as approaches is sometimes denoted by a right arrow (→ or \rightarrow), as in :f(x) \to L \text x \to c, which reads "f of x tends to L as x tends to c". History Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a geometric series in his work ''Opus Geometricum'' (1647): "The ''terminus'' of a pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite Product
In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product : \prod_^ a_n = a_1 a_2 a_3 \cdots is defined to be the limit of a sequence, limit of the Multiplication#Capital pi notation, partial products ''a''1''a''2...''a''''n'' as ''n'' increases without bound. The product is said to ''Limit of a sequence, converge'' when the limit exists and is not zero. Otherwise the product is said to ''diverge''. A limit of zero is treated specially in order to obtain results analogous to those for Infinite series, infinite sums. Some sources allow convergence to 0 if there are only a finite number of zero factors and the product of the non-zero factors is non-zero, but for simplicity we will not allow that here. If the product converges, then the limit of the sequence ''a''''n'' as ''n'' increases without bound must be 1, while the converse is in general not true. The best known examples of infinite products are probably some of the formulae for pi, &p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Analysis
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as becomes very large, the term becomes insignificant compared to . The function is said to be "''asymptotically equivalent'' to , as ". This is often written symbolically as , which is read as " is asymptotic to ". An example of an important asymptotic result is the prime number theorem. Let denote the prime-counting function (which is not directly related to the constant pi), i.e. is the number of prime numbers that are less than or equal to . Then the theorem states that \pi(x)\sim\frac. Asymptotic analysis is commonly used in computer science as part of the analysis of algorithms and is often expressed there in terms of big O notation. Definition Formally, given functions and , we define a binary relation f(x) \sim g(x) \qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description. The sequence 0, 3, 8, 15, ... is formed according to the formula ''n''2 − 1 for the ''n''th term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, even though we do not have a formula for the ''n''th perfect number. Examples Integer sequences that have their own name include: *Abundant numbers *Baum–Sweet sequence *Bell numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of several real variables. In this generalization, the derivativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lerch Transcendent
In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about the function in 1887. Definition The Lerch zeta function is given by :L(\lambda, s, \alpha) = \sum_^\infty \frac . A related function, the Lerch transcendent, is given by :\Phi(z, s, \alpha) = \sum_^\infty \frac . The two are related, as :\,\Phi(e^, s,\alpha)=L(\lambda, s, \alpha). Integral representations The Lerch transcendent has an integral representation: : \Phi(z,s,a)=\frac\int_0^\infty \frac\,dt The proof is based on using the integral definition of the Gamma function to write :\Phi(z,s,a)\Gamma(s) = \sum_^\infty \frac \int_0^\infty x^s e^ \frac = \sum_^\infty \int_0^\infty t^s z^n e^ \frac and then interchanging the sum and integral. The resulting integral representation converges for z \in \Complex \setm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the base is implicit, simply . Parentheses are sometimes added for clarity, giving , , or . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity. The natural logarithm of is the power to which would have to be raised to equal . For example, is , because . The natural logarithm of itself, , is , because , while the natural logarithm of is , since . The natural logarithm can be defined for any positive real number as the area under the curve from to (with the area being negative when ). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural". The definition of the natural logarithm can then b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jonathan Sondow
Jonathan may refer to: *Jonathan (name), a masculine given name Media * ''Jonathan'' (1970 film), a German film directed by Hans W. Geißendörfer * ''Jonathan'' (2016 film), a German film directed by Piotr J. Lewandowski * ''Jonathan'' (2018 film), an American film directed by Bill Oliver * ''Jonathan'' (Buffy comic), a 2001 comic book based on the ''Buffy the Vampire Slayer'' television series * ''Jonathan'' (TV show), a Welsh-language television show hosted by ex-rugby player Jonathan Davies People and biblical figures Bible *Jonathan (1 Samuel), son of King Saul of Israel and friend of David, in the Books of Samuel *Jonathan (Judges), in the Book of Judges Judaism *Jonathan Apphus, fifth son of Mattathias and leader of the Hasmonean dynasty of Judea from 161 to 143 BCE *Rabbi Jonathan, 2nd century *Jonathan (High Priest), a High Priest of Israel in the 1st century Other *Jonathan (apple), a variety of apple * "Jonathan" (song), a 2015 song by French singer and songwrite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lerch's Transcendent
In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about the function in 1887. Definition The Lerch zeta function is given by :L(\lambda, s, \alpha) = \sum_^\infty \frac . A related function, the Lerch transcendent, is given by :\Phi(z, s, \alpha) = \sum_^\infty \frac . The two are related, as :\,\Phi(e^, s,\alpha)=L(\lambda, s, \alpha). Integral representations The Lerch transcendent has an integral representation: : \Phi(z,s,a)=\frac\int_0^\infty \frac\,dt The proof is based on using the integral definition of the Gamma function to write :\Phi(z,s,a)\Gamma(s) = \sum_^\infty \frac \int_0^\infty x^s e^ \frac = \sum_^\infty \int_0^\infty t^s z^n e^ \frac and then interchanging the sum and integral. The resulting integral representation converges for z \in \Complex \setm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
