BUILT-IN MATHEMATICA SYMBOL

BernoulliB

BernoulliB[n]
gives the Bernoulli number

.

BernoulliB

gives the Bernoulli polynomial

.

MORE INFORMATION

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.

EXAMPLES

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Basic Examples (2)

First 10 Bernoulli numbers:

Bernoulli polynomials:

First 10 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

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 (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,

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: