Bernoulli number en · NOUN
Etymology
Named after Swiss mathematician Jacob Bernoulli (1654–1705), who discovered the numbers independently of and at about the same time as Japanese mathematician Seki Kōwa. The numbers first appeared as coefficients in Bernoulli's formulae for the sum of the first n positive integers, each raised to a given power. The sequence was subsequently found in numerous other contexts.
Meanings
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Any one of the rational numbers in a sequence of such that appears in numerous contexts, including formulae for sums of integer powers and certain power series expansions.
1993, Serge Lang, Complex Analysis, Springer, 3rd Edition, page 418, The assertion about the value of the zeta function at negative integers then comes immediately from the definition of the Bernoulli numbers in terms of the coefficients of a power series, namely t/(eᵗ-1)=∑ₙ₌₀ ᪲B_n(tⁿ)/(n!)
This done, he extends occurrence of Bernoulli numbers in the expansion of #92;textstyle#92;frac 1 2#92;cot(#92;frac 1 2x) to the more general form #92;textstyle#92;frac#92;pin#92;cot(#92;frac#123;m#92;pi#125;n) and uses that to relate Bernoulli numbers to the values of #92;zeta(2n). To end the theoretical parts of his exposition, he gives some of the properties of the Bernoulli polynomials and notes that Bernoulli numbers grow faster than any geometric series.
2007, C. Edward Sandifer, How Euler Did It, Mathematical Association of America, page 175:In the first three sections, we present standard results about the Bernoulli numbers and polynomials and the Riemann zeta functions and its zeros.
2002, M. Aslam Chaudhry, Syed M. Zubair, On a Class of Incomplete Gamma Functions with Applications, CRC Press (Chapman & Hall/CRC), page 287:
Forms
| Spelling | Features | Labels | Source |
|---|---|---|---|
| Bernoulli's number | alternative | lexicographic | |
| Bernoulli numbers | Number=Plur | lexicographic |