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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 .
  • BernoulliB can be evaluated to arbitrary numerical precision.
First 10 Bernoulli numbers:
Bernoulli polynomials:
First 10 Bernoulli numbers:
Click for copyable input
Bernoulli polynomials:
Click for copyable input
BernoulliB threads element-wise over lists:
Plot Bernoulli polynomials:
TraditionalForm formatting:
Find sums of powers ∑_(k=1)^nk^j 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:
Find BernoulliB numbers from their generating function:
Find BernoulliB polynomials from their generating function:
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:
Going from Bernoulli numbers to Bernoulli polynomials with umbral calculus:
The 20000^(th) Bernoulli number can be computed in under a second:
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