[size=150]A large number of common functions have power-series expansions whose coefficients can be expressed in terms of a set of numbers first identified by Jacques Bernoulli. These so-called [b]Bernoulli numbers[/b] also appear in other mathematical contexts; they are found in formulas for the Riemann zeta functions of even integer argument, and they enter in the Euler-Maclaurin summation formula (which relates the integral of a function to a sum over discrete values of the same function). They are also of interest from a pedagogical viewpoint; their definition and properties illustrate the concept and use of [b]generating functions[/b] and recurrence formulas.[br][br]The Bernoulli numbers [img width=15,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline1.gif[/img] are a sequence of signed rational numbers that can be defined by the [url=https://mathworld.wolfram.com/ExponentialGeneratingFunction.html]exponential generating function[/url][/size][table][tr][td][img width=113,height=44]https://mathworld.wolfram.com/images/equations/BernoulliNumber/NumberedEquation1.gif[/img][/td][td][br][/td][/tr][/table][size=150]These numbers arise in the series expansions of trigonometric functions, and are extremely important in [url=https://mathworld.wolfram.com/NumberTheory.html]number theory[/url] and [url=https://mathworld.wolfram.com/Analysis.html]analysis[/url]. There are actually two definitions for the Bernoulli numbers. To distinguish them, the Bernoulli numbers as defined in modern usage (National Institute of Standards and Technology convention) are written [img width=15,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline2.gif[/img], while the Bernoulli numbers encountered in older literature are written [img width=15,height=17]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline3.gif[/img] (Gradshteyn and Ryzhik 2000). In each case, the Bernoulli numbers are a special case of the [url=https://mathworld.wolfram.com/BernoulliPolynomial.html]Bernoulli polynomials[/url] [img width=33,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline4.gif[/img] or [img width=33,height=17]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline5.gif[/img] with [img width=65,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline6.gif[/img] and [img width=65,height=17]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline7.gif[/img].[br][br]The Bernoulli number and polynomial should not be confused with the [url=https://mathworld.wolfram.com/BellNumber.html]Bell numbers[/url] and [url=https://mathworld.wolfram.com/BellPolynomial.html]Bell polynomial[/url], which are also commonly denoted [img width=15,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline8.gif[/img] and [img width=33,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline9.gif[/img], respectively. Bernoulli numbers defined by the modern definition are denoted [img width=15,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline10.gif[/img] and sometimes called "even-index" Bernoulli numbers. These are the Bernoulli numbers returned, by example, by the [url=http://www.wolfram.com/language/]Wolfram Language[/url] function [url=http://reference.wolfram.com/language/ref/BernoulliB.html]Bernoulli B[/url][[i]n[/i]].The Bernoulli number [img width=15,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline11.gif[/img] can be defined by the [url=https://mathworld.wolfram.com/ContourIntegral.html]contour integral[br][/url][/size][table][tr][td][img width=149,height=37]https://mathworld.wolfram.com/images/equations/BernoulliNumber/NumberedEquation2.gif[/img][br][br][/td][td][/td][/tr][/table]w[size=150]here the contour encloses the origin, has a radius less than [img width=17,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline12.gif[/img] (to avoid the poles at [img width=34,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline13.gif[/img]), and is traversed in a counterclockwise direction (Arfken 1985, p. 413).The first few Bernoulli numbers [img width=15,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline14.gif[/img] are:[/size][table][tr][td][img width=15,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline15.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline16.gif[/img][/td][td][img width=7,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline17.gif[/img][/td][td][/td][/tr][tr][td][img width=15,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline18.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline19.gif[/img][/td][td][img width=20,height=23]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline20.gif[/img][/td][td][/td][/tr][tr][td][img width=15,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline21.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline22.gif[/img][/td][td][img width=10,height=23]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline23.gif[/img][/td][td][/td][/tr][tr][td][img width=15,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline24.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline25.gif[/img][/td][td][img width=25,height=23]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline26.gif[/img][/td][td][/td][/tr][tr][td][img width=15,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline27.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline28.gif[/img][/td][td][img width=15,height=23]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline29.gif[/img][/td][td][/td][/tr][tr][td][img width=15,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline30.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline31.gif[/img][/td][td][img width=25,height=23]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline32.gif[/img][/td][td][/td][/tr][tr][td][img width=20,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline33.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline34.gif[/img][/td][td][img width=15,height=23]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline35.gif[/img][/td][td][/td][/tr][tr][td][img width=20,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline36.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline37.gif[/img][/td][td][img width=35,height=23]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline38.gif[/img][/td][td][/td][/tr][tr][td][img width=20,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline39.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline40.gif[/img][/td][td][img width=10,height=23]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline41.gif[/img][/td][td][/td][/tr][tr][td][img width=20,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline42.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline43.gif[/img][/td][td][img width=35,height=23]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline44.gif[/img][/td][td][/td][/tr][tr][td][img width=20,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline45.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline46.gif[/img][/td][td][img width=31,height=23]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline47.gif[/img][/td][td][/td][/tr][tr][td][img width=20,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline48.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline49.gif[/img][/td][td][img width=46,height=23]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline50.gif[/img][/td][td][/td][/tr][tr][td][img width=20,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline51.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline52.gif[/img][/td][td][img width=36,height=23]https://mathworld.wolfram.com/images/equations/BernoulliNumber/Inline53.gif[/img][/td][/tr][/table]