- Mathematical function, suitable for both symbolic and numerical manipulation.
- The Bernoulli polynomials satisfy the generating function relation .
- The Bernoulli numbers are given by .
- BernoulliB can be evaluated to arbitrary numerical precision.
- BernoulliB automatically threads over lists.
Examplesopen allclose all
Basic Examples (2)
First 10 Bernoulli numbers:
Find sums of powers using BernoulliB:
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)
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 (2)
Going from Bernoulli numbers to Bernoulli polynomials with umbral calculus:
The 20000 Bernoulli number can be computed in under a second:
Introduced in 1988
Updated in 2008