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Wallis Product
In mathematics, the Wallis product for , published in 1656 by John Wallis, states that :\begin \frac & = \prod_^ \frac = \prod_^ \left(\frac \cdot \frac\right) \\ pt& = \Big(\frac \cdot \frac\Big) \cdot \Big(\frac \cdot \frac\Big) \cdot \Big(\frac \cdot \frac\Big) \cdot \Big(\frac \cdot \frac\Big) \cdot \; \cdots \\ \end Proof using integration Wallis derived this infinite product as it is done in calculus books today, by examining \int_0^\pi \sin^n x\,dx for even and odd values of n, and noting that for large n, increasing n by 1 results in a change that becomes ever smaller as n increases. Let :I(n) = \int_0^\pi \sin^n x\,dx. (This is a form of Wallis' integrals.) Integrate by parts: :\begin u &= \sin^x \\ \Rightarrow du &= (n-1) \sin^x \cos x\,dx \\ dv &= \sin x\,dx \\ \Rightarrow v &= -\cos x \end :\begin \Rightarrow I(n) &= \int_0^\pi \sin^n x\,dx \\ pt &= -\sin^x\cos x \Biggl, _0^\pi - \int_0^\pi (-\cos x)(n-1) \sin ... [...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] |
Dirichlet Eta Function
In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: \eta(s) = \sum_^ = \frac - \frac + \frac - \frac + \cdots\approx \prod_^ \infty . This Dirichlet series is the alternating sum corresponding to the Dirichlet series expansion of the Riemann zeta function, ''ζ''(''s'') — and for this reason the Dirichlet eta function is also known as the alternating zeta function, also denoted ''ζ''*(''s''). The following relation holds: \eta(s) = \left(1-2^\right) \zeta(s) Both Dirichlet eta function and Riemann zeta function are special cases of Polylogarithm. While the Dirichlet series expansion for the eta function is convergent only for any complex number ''s'' with real part > 0, it is Abel summable for any complex number. This serves to define the eta function as an entire function. (The above relation and the facts that the eta function is ent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Articles Containing Proofs
Article often refers to: * Article (grammar), a grammatical element used to indicate definiteness or indefiniteness * Article (publishing), a piece of nonfictional prose that is an independent part of a publication Article may also refer to: Government and law * Article (European Union), articles of treaties of the European Union * Articles of association, the regulations governing a company, used in India, the UK and other countries * Articles of clerkship, the contract accepted to become an articled clerk * Articles of Confederation, the predecessor to the current United States Constitution *Article of Impeachment, a formal document and charge used for impeachment in the United States * Articles of incorporation, for corporations, U.S. equivalent of articles of association * Articles of organization, for limited liability organizations, a U.S. equivalent of articles of association Other uses * Article, an HTML element, delimited by the tags and * Article of clothing, an ite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3Blue1Brown
3Blue1Brown is a math YouTube channel created and run by Grant Sanderson. The channel focuses on teaching higher mathematics from a visual perspective, and on the process of discovery and inquiry-based learning in mathematics, which Sanderson calls "inventing math". , the channel has 4.89 million subscribers. Early life and education Sanderson graduated from Stanford University in 2015 with a bachelor's degree in math. He worked for Khan Academy from 2015 to 2016 as part of their content fellowship program, producing videos and articles about multivariable calculus, after which he started focusing his full attention on 3Blue1Brown. Career 3Blue1Brown started as a personal programming project in early 2015. In a podcast of ''Showmakers'', Sanderson explained that he wanted to practice his coding skills and decided to make a graphics library in Python, which eventually became the open-source project "Manim" (short for Mathematical Animation engine). To have a goal for the project ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
YouTube
YouTube is a global online video platform, online video sharing and social media, social media platform headquartered in San Bruno, California. It was launched on February 14, 2005, by Steve Chen, Chad Hurley, and Jawed Karim. It is owned by Google, and is the List of most visited websites, second most visited website, after Google Search. YouTube has more than 2.5 billion monthly users who collectively watch more than one billion hours of videos each day. , videos were being uploaded at a rate of more than 500 hours of content per minute. In October 2006, YouTube was bought by Google for $1.65 billion. Google's ownership of YouTube expanded the site's business model, expanding from generating revenue from advertisements alone, to offering paid content such as movies and exclusive content produced by YouTube. It also offers YouTube Premium, a paid subscription option for watching content without ads. YouTube also approved creators to participate in Google's Google AdSens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wallis Sieve
Wallis (derived from ''Wallace'') may refer to: People * Wallis (given name) **Wallis, Duchess of Windsor * Wallis (surname) Places * Wallis (Ambleston), a hamlet within the parish of Ambleston in Pembrokeshire, West Wales, United Kingdom * Wallis, Mississippi, an unincorporated community, United States * Wallis, Texas, a city, United States * Wallis and Futuna, a French overseas department ** Wallis Island, one of the islands of Wallis and Futuna * Valais, a Swiss canton with the German name "Wallis" * Walliswil bei Niederbipp, a municipality in the Oberaargau administrative district, canton of Bern, Switzerland * Walliswil bei Wangen, a municipality in the Oberaargau administrative district, canton of Bern, Switzerland Brands and enterprises * Wallis (retailer), a British clothing retailer * Wallis Theatres, an Australian cinema franchise See also * Wallace (other) Wallace may refer to: People * Clan Wallace in Scotland * Wallace (given name) * Wallace (su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Product
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Euler. This series and its continuation to the entire complex plane would later become known as the Riemann zeta function. Definition In general, if is a bounded multiplicative function, then the Dirichlet series :\sum_ \frac\, is equal to :\prod_ P(p, s) \quad \text \operatorname(s) >1 . where the product is taken over prime numbers , and is the sum :\sum_^\infty \frac = 1 + \frac + \frac + \frac + \cdots In fact, if we consider these as formal generating functions, the existence of such a ''formal'' Euler product expansion is a necessary and sufficient condition that be multiplicative: this says exactly that is the product of the whenever factors as the product of the powers of distinct primes . An important special case is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leibniz Formula For π
In mathematics, the Leibniz formula for , named after Gottfried Leibniz, states that 1-\frac+\frac-\frac+\frac-\cdots=\frac, an alternating series. It is also called the 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. The series for the inverse tangent function, which is also known as Gregory's series, can be given by: : \arctan x = x - \frac + \frac - \frac + \cdots The Leibniz formula for \frac can be obtained by putting x=1 into this series. It also is the Dirichlet -series of the non-principal Dirichlet character of modulus 4 evaluated at s=1, and, therefore, the value of the Dirichlet beta function. Proofs Proof 1 \begin \frac &= \arctan(1) \\ &= \int_0^1 \frac 1 \, dx \\ pt& = \int_0^1\left(\sum_^n (-1)^k x^+\frac\right) \, dx \\ pt& = \left(\sum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Viète's Formula
In mathematics, Viète's formula is the following infinite product of nested radicals representing twice the reciprocal of the mathematical constant : \frac2\pi = \frac2 \cdot \frac2 \cdot \frac2 \cdots It can also be represented as: \frac2\pi = \prod_^ \cos \frac The formula is named after François Viète, who published it in 1593. As the first formula of European mathematics to represent an infinite process, it can be given a rigorous meaning as a limit expression, and marks the beginning of mathematical analysis. It has linear convergence, and can be used for calculations of , but other methods before and since have led to greater accuracy. It has also been used in calculations of the behavior of systems of springs and masses, and as a motivating example for the concept of statistical independence. The formula can be derived as a telescoping product of either the areas or perimeters of nested polygons converging to a circle. Alternatively, repeated use of the half-angle fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinitesimal Calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", 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. It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Later work, including codifying the idea of limits, put these developments on a more solid conceptual footing. Today, calculus has widespread uses in scien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Zeta Function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > 1 and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory, and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that is consid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
