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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'' (Петербургская Академия наук). [1]

46 relations: Akiva Yaglom, Apéry's constant, Augustin-Louis Cauchy, Basel, Bernhard Riemann, Bernoulli family, Bernoulli number, Binomial theorem, Calculus, Closed-form expression, Complex analysis, Complex number, De Moivre's formula, Fourier analysis, Fourier series, Isaak Yaglom, Leonhard Euler, Limit of a function, Mathematical analysis, Mathematical Association of America, Mathematical proof, Mathematician, Multiplicative inverse, Multivariable calculus, Natural number, Newton's identities, Number theory, On the Number of Primes Less Than a Given Magnitude, Parseval's identity, Peter Swinnerton-Dyer, Pietro Mengoli, Polynomial, Prime number, Proofs from THE BOOK, Residue theorem, Riemann zeta function, Series (mathematics), Springer Science+Business Media, Square number, Squeeze theorem, Summation, Taylor series, Trigonometric functions, University of Cambridge, Vieta's formulas, Weierstrass factorization theorem.

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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Apéry's constant

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

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Augustin-Louis Cauchy

Baron Augustin-Louis Cauchy FRS FRSE (21 August 1789 – 23 May 1857) was a French mathematician reputed as a pioneer of analysis.

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Basel

Basel (or; or less often used Basle; Basel; Bâle; Basilea; Basilea) is Switzerland's third most populous city (behind Zürich and Geneva) with about 195,000 inhabitants.

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Bernhard Riemann

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

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Bernoulli family

The Bernoullis were a patrician family of merchants and scholars, originally from Antwerp, who resettled in Basel, Switzerland.

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Bernoulli number

In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to 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

Calculus is the mathematical study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations.

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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 the imaginary unit, that satisfies 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) x and integer n it holds that where i is the imaginary unit (i2.

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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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Fourier series

In mathematics, a Fourier series is a way to represent a (wave-like) function as the sum of simple sine waves.

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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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Leonhard Euler

Leonhard Euler (17071783) was a pioneering Swiss mathematician and physicist.

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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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Mathematical analysis

Mathematical analysis is a branch of mathematics that studies continuous change and includes the theories of differentiation, integration, measure, limits, 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 proof

In mathematics, a proof is a deductive argument for a mathematical statement.

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Mathematician

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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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 in more than one variable: the differentiation and integration of functions involving multiple variables, rather than just one.

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Natural number

In mathematics, the natural numbers (sometimes called the whole numbers): "whole number An integer, though sometimes it is taken to mean only non-negative integers, or just the positive integers." give definitions of "whole number" under several headwords: INTEGER … Syn. whole number.

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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 arithmeticEspecially in older sources; see two following notes.) 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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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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Peter Swinnerton-Dyer

Sir Henry Peter Francis Swinnerton-Dyer, 16th Baronet KBE FRS (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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Polynomial

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

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Prime number

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.

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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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Residue theorem

In complex analysis, a field in mathematics, the residue theorem, sometimes called Cauchy's residue theorem (one of many things named after Augustin-Louis Cauchy), 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, ζ(s), is a function of a complex variable s that analytically continues the sum of the infinite series which converges when the real part of s is greater than 1.

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Series (mathematics)

A series is, informally speaking, the sum of the terms of a sequence.

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Springer Science+Business Media

Springer Science+Business Media or Springer is a global publishing company that publishes books, e-books and peer-reviewed journals in science, 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 known also 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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Summation

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

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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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Trigonometric functions

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

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University of Cambridge

The University of CambridgeThe corporate title of the university is The Chancellor, Masters, and Scholars of the University of Cambridge.

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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).

References

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

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