gives the Bernoulli number .
gives the Bernoulli polynomial .
- Mathematical function, suitable for both symbolic and numerical manipulation.
- The Bernoulli polynomials satisfy the generating function relation .
- The Bernoulli numbers are given by .
- For odd , the Bernoulli numbers are equal to 0, except .
- BernoulliB can be evaluated to arbitrary numerical precision.
- BernoulliB automatically threads over lists.
Examplesopen allclose all
BernoulliB threads element-wise over lists:
Find sums of powers using BernoulliB:
Compare with direct summation:
Set up an Euler–Maclaurin integration formula:
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:
Wolfram Research (1988), BernoulliB, Wolfram Language function, https://reference.wolfram.com/language/ref/BernoulliB.html (updated 2008).
Wolfram Language. 1988. "BernoulliB." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2008. https://reference.wolfram.com/language/ref/BernoulliB.html.
Wolfram Language. (1988). BernoulliB. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BernoulliB.html