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Basel problem

Index Basel problem

The Basel problem is a problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1734 and read on 5 December 1735 in ''The Saint Petersburg Academy of Sciences''. [1]

71 relations: Akiva Yaglom, Almost integer, Apéry's constant, Augustin-Louis Cauchy, Basel, Bernhard Riemann, Bernoulli family, Bernoulli number, Binomial theorem, Calculus, Cauchy product, Closed-form expression, Complex analysis, Complex number, De Moivre's formula, Elementary symmetric polynomial, Euler–Mascheroni constant, Fourier analysis, Generating function, Heuristic, Hilbert space, Injective function, Inner product space, Integration by parts, Irrationality, Isaak Yaglom, Karl Weierstrass, Leonhard Euler, Limit of a function, List of trigonometric identities, List of zeta functions, Logarithm, Mathematical analysis, Mathematical Association of America, Mathematical induction, Mathematical proof, Mathematician, Mathematics, Multiplicative inverse, Multivariable calculus, Natural number, Newton's identities, Number theory, On the Number of Primes Less Than a Given Magnitude, Orthonormal basis, Parseval's identity, Particular values of the Riemann zeta function, Periodic function, Peter Swinnerton-Dyer, Pietro Mengoli, ..., Polynomial, Prime number, Proofs from THE BOOK, Rationality, Recurrence relation, Residue theorem, Riemann zeta function, Series (mathematics), Springer Science+Business Media, Square number, Squeeze theorem, Summation, Sums of powers, Taylor series, The Mathematical Intelligencer, Transcendental number, Trigonometric functions, University of Cambridge, Upper and lower bounds, Vieta's formulas, Weierstrass factorization theorem. Expand index (21 more) »

Akiva Yaglom

Akiva Moiseevich Yaglom (Аки́ва Моисе́евич Ягло́м; 6 March 1921 – 13 December 2007) was a Soviet and Jewish physicist, mathematician, statistician, and meteorologist.

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Almost integer

In recreational mathematics, an almost integer (or near-integer) is any number that is not an integer but is very close to one.

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Apéry's constant

In mathematics, at the intersection of number theory and special functions, Apéry's constant is defined as the number where is the Riemann zeta function.

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Augustin-Louis Cauchy

Baron Augustin-Louis Cauchy FRS FRSE (21 August 178923 May 1857) was a French mathematician, engineer and physicist who made pioneering contributions to several branches of mathematics, including: mathematical analysis and continuum mechanics.

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Basel (also Basle; Basel; Bâle; Basilea) is a city in northwestern Switzerland on the river Rhine.

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Bernhard Riemann

Georg Friedrich Bernhard Riemann (17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.

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Bernoulli family

The Bernoulli family of Basel is a patrician family, notable for having produced eight mathematically gifted academics who, between them, contributed to the foundations of applied mathematics and physics during the early modern period.

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Bernoulli number

In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in number theory.

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Binomial theorem

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.

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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.

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Cauchy product

In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series.

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Closed-form expression

In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations.

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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.

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Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

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De Moivre's formula

In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity), named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) and integer it holds that where is the imaginary unit.

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Elementary symmetric polynomial

In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symmetric polynomials.

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Euler–Mascheroni constant

The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma.

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Fourier analysis

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.

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Generating function

In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a power series.

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A heuristic technique (εὑρίσκω, "find" or "discover"), often called simply a heuristic, is any approach to problem solving, learning, or discovery that employs a practical method, not guaranteed to be optimal, perfect, logical, or rational, but instead sufficient for reaching an immediate goal.

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Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

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Injective function

In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.

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Inner product space

In linear algebra, an inner product space is a vector space with an additional structure called an inner product.

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Integration by parts

In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of their derivative and antiderivative.

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Irrationality is cognition, thinking, talking, or acting without inclusion of rationality.

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Isaak Yaglom

Isaak Moiseevich Yaglom (Исаа́к Моисе́евич Ягло́м; 6 March 1921 – 17 April 1988) was a Soviet mathematician and author of popular mathematics books, some with his twin Akiva Yaglom.

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Karl Weierstrass

Karl Theodor Wilhelm Weierstrass (Weierstraß; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis".

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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.

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Limit of a function

Although the function (sin x)/x is not defined at zero, as x becomes closer and closer to zero, (sin x)/x becomes arbitrarily close to 1.

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List of trigonometric identities

In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined.

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List of zeta functions

In mathematics, a zeta function is (usually) a function analogous to the original example: the Riemann zeta function Zeta functions include.

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In mathematics, the logarithm is the inverse function to exponentiation.

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Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

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Mathematical Association of America

The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level.

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Mathematical induction

Mathematical induction is a mathematical proof technique.

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Mathematical proof

In mathematics, a proof is an inferential argument for a mathematical statement.

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A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

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Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1.

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Multivariable calculus

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables, rather than just one.

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Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

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Newton's identities

In mathematics, Newton's identities, also known as the Newton–Girard formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials.

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Number theory

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

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On the Number of Primes Less Than a Given Magnitude

" die Anzahl der Primzahlen unter einer gegebenen " (usual English translation: "On the Number of Primes Less Than a Given Magnitude") is a seminal 10-page paper by Bernhard Riemann published in the November 1859 edition of the Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin.

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Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.

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Parseval's identity

In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function.

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Particular values of the Riemann zeta function

This article gives some specific values of the Riemann zeta function, including values at integer arguments, and some series involving them.

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Periodic function

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods.

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Peter Swinnerton-Dyer

Sir Henry Peter Francis Swinnerton-Dyer, 16th Baronet (born 2 August 1927), commonly known as Peter Swinnerton-Dyer, is an English mathematician specialising in number theory at University of Cambridge.

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Pietro Mengoli

Pietro Mengoli (1626, Bologna – June 7, 1686, Bologna) was an Italian mathematician and clergyman from Bologna, where he studied with Bonaventura Cavalieri at the University of Bologna, and succeeded him in 1647.

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In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

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Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

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Proofs from THE BOOK

Proofs from THE BOOK is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler.

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Rationality is the quality or state of being rational – that is, being based on or agreeable to reason.

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Recurrence relation

In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms.

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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.

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Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function,, is a function of a complex variable s that analytically continues the sum of the Dirichlet series which converges when the real part of is greater than 1.

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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.

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Springer Science+Business Media

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

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Square number

In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself.

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Squeeze theorem

In calculus, the squeeze theorem, also known as the pinching theorem, the sandwich theorem, the sandwich rule, and sometimes the squeeze lemma, is a theorem regarding the limit of a function.

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In mathematics, summation (capital Greek sigma symbol: ∑) is the addition of a sequence of numbers; the result is their sum or total.

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Sums of powers

In mathematics and statistics, sums of powers occur in a number of contexts.

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Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

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The Mathematical Intelligencer

The Mathematical Intelligencer is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals.

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Transcendental number

In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients.

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Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.

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University of Cambridge

The University of Cambridge (informally Cambridge University)The corporate title of the university is The Chancellor, Masters, and Scholars of the University of Cambridge.

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Upper and lower bounds

In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set (K, ≤) is an element of K which is greater than or equal to every element of S. The term lower bound is defined dually as an element of K which is less than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound.

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Vieta's formulas

In mathematics, Vieta's formulas are formulas that relate the coefficients of a polynomial to sums and products of its roots.

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Weierstrass factorization theorem

In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that entire functions can be represented by a product involving their zeroes.

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1 + 1/4 + 1/9 + 1/16 + · · ·, Basel series, Basel sum, Evaluation of z(2), Evaluation of ζ(2), Riemann zeta function zeta(2), Series of reciprocal squares, Zeta(2).


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

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