BernoulliB

BernoulliB[n]

gives the Bernoulli number TemplateBox[{n}, BernoulliB].

BernoulliB[n,x]

gives the Bernoulli polynomial TemplateBox[{n, x}, BernoulliB2].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The Bernoulli polynomials satisfy the generating function relation te^(xt)/(e^t-1)=sum_(n=0)^(infty)TemplateBox[{n, x}, BernoulliB2](t^n/n!).
  • The Bernoulli numbers are given by TemplateBox[{n}, BernoulliB]=TemplateBox[{n, 0}, BernoulliB2].
  • For odd , the Bernoulli numbers are equal to 0, except TemplateBox[{1}, BernoulliB]=-1/2.
  • BernoulliB can be evaluated to arbitrary numerical precision.
  • BernoulliB automatically threads over lists.

Examples

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Basic Examples  (2)

First 10 Bernoulli numbers:

Bernoulli polynomials:

Scope  (3)

BernoulliB threads element-wise over lists:

Plot Bernoulli polynomials:

TraditionalForm formatting:

Applications  (6)

Find sums of powers using BernoulliB (Faulhaber's formula):

Compare with direct summation:

Set up an Euler–Maclaurin integration formula:

Use it for :

Compare with the exact summation result:

Plot roots of Bernoulli polynomials in the complex plane:

Show the approach of Bernoulli numbers to a limiting form:

The denominator of Bernoulli numbers is given by the von Staudt–Clausen formula:

Compute Bernoulli numbers in modular arithmetic modulo a prime:

Properties & Relations  (3)

Find BernoulliB numbers from their generating function:

Find BernoulliB polynomials from their generating function:

BernoulliB can be represented as a DifferenceRoot:

Possible Issues  (2)

Algorithmically produced results are frequently expressed using Zeta instead of BernoulliB:

When entered in the traditional form, is not automatically interpreted as a Bernoulli number:

Neat Examples  (3)

Going from Bernoulli numbers to Bernoulli polynomials with umbral calculus:

The 20000^(th) Bernoulli number can be computed in under a second:

Define a Hankel matrix whose entries are the Bernoulli numbers:

Its determinant can be expressed in terms of the Barnes G-function:

Wolfram Research (1988), BernoulliB, Wolfram Language function, https://reference.wolfram.com/language/ref/BernoulliB.html (updated 2008).

Text

Wolfram Research (1988), BernoulliB, Wolfram Language function, https://reference.wolfram.com/language/ref/BernoulliB.html (updated 2008).

CMS

Wolfram Language. 1988. "BernoulliB." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2008. https://reference.wolfram.com/language/ref/BernoulliB.html.

APA

Wolfram Language. (1988). BernoulliB. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BernoulliB.html

BibTeX

@misc{reference.wolfram_2024_bernoullib, author="Wolfram Research", title="{BernoulliB}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/BernoulliB.html}", note=[Accessed: 22-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_bernoullib, organization={Wolfram Research}, title={BernoulliB}, year={2008}, url={https://reference.wolfram.com/language/ref/BernoulliB.html}, note=[Accessed: 22-November-2024 ]}