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BernoulliB

BernoulliB[n]
gives the Bernoulli number B_n.
BernoulliB[n, x]
gives the Bernoulli polynomial B_n(x).
  • 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)B_n(x)(t^n/n!).
  • The Bernoulli numbers are given by B_n=B_n(0).
  • BernoulliB can be evaluated to arbitrary numerical precision.
EXAMPLESCLOSE ALL
Basic Examples   (2)
First ten Bernoulli numbers:
Bernoulli polynomials:
First ten Bernoulli numbers:
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Bernoulli polynomials:
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Scope   (3)
BernoulliB threads element-wise over lists:
Plot Bernoulli polynomials:
TraditionalForm formatting:
Applications   (6)
Find sums of powers sum_(k=1)^nk^j using BernoulliB:
Compare with direct summation:
Set up an Euler-Maclaurin integration formula:
Use it for f(x)=x^(5/3):
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 arithmeric modulo a prime:
Properties & Relations   (2)
Find BernoulliB numbers from their generating function:
Find BernoulliB polynomials from their generating function:
Possible Issues   (2)
Algorithmically produced results are frequently expressed using Zeta instead of BernoulliB:
When entered in the traditional form, B_n is not automatically interpreted as a Bernoulli number:
Neat Examples   (2)
Going from Bernoulli numbers to Bernoulli polynomials with umbral calculus:
20000-th Bernoulli number can be computed in under a second:
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