# BernoulliB

BernoulliB[n]

gives the Bernoulli number .

BernoulliB[n,x]

gives the Bernoulli polynomial .

# Details • 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:

## Scope(3)

Plot Bernoulli polynomials:

## Applications(6)

Find sums of powers using BernoulliB:

Compare with direct summation:

Set up an EulerMaclaurin 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 StaudtClausen 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:

## Neat Examples(2)

Going from Bernoulli numbers to Bernoulli polynomials with umbral calculus:

The 20000 Bernoulli number can be computed in under a second: