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bernoulli

Bernoulli numbers and polynomials

Description

example

bernoulli(n) returns the nth Bernoulli number.

example

bernoulli(n,x) returns the nth Bernoulli polynomial.

Examples

Bernoulli Numbers with Odd and Even Indices

The 0th Bernoulli number is 1. The next Bernoulli number can be -1/2 or 1/2, depending on the definition. The bernoulli function uses -1/2. The Bernoulli numbers with even indices n > 1 alternate the signs. Any Bernoulli number with an odd index n > 2 is 0.

Compute the even-indexed Bernoulli numbers with the indices from 0 to 10. Because these indices are not symbolic objects, bernoulli returns floating-point results.

bernoulli(0:2:10)
ans =
1.0000    0.1667   -0.0333    0.0238   -0.0333    0.0758

Compute the same Bernoulli numbers for the indices converted to symbolic objects:

bernoulli(sym(0:2:10))
ans =
[ 1, 1/6, -1/30, 1/42, -1/30, 5/66]

Compute the odd-indexed Bernoulli numbers with the indices from 1 to 11:

bernoulli(sym(1:2:11))
ans =
[ -1/2, 0, 0, 0, 0, 0]

Bernoulli Polynomials

For the Bernoulli polynomials, use bernoulli with two input arguments.

Compute the first, second, and third Bernoulli polynomials in variables x, y, and z, respectively:

syms x y z
bernoulli(1, x)
bernoulli(2, y)
bernoulli(3, z)
ans =
x - 1/2

ans =
y^2 - y + 1/6

ans =
z^3 - (3*z^2)/2 + z/2

If the second argument is a number, bernoulli evaluates the polynomial at that number. Here, the result is a floating-point number because the input arguments are not symbolic numbers:

bernoulli(2, 1/3)
ans =
-0.0556

To get the exact symbolic result, convert at least one of the numbers to a symbolic object:

bernoulli(2, sym(1/3))
ans =
-1/18

Plot Bernoulli Polynomials

Plot the first six Bernoulli polynomials.

syms x
fplot(bernoulli(0:5, x), [-0.8 1.8])
title('Bernoulli Polynomials')
grid on Handle Expressions Containing Bernoulli Polynomials

Many functions, such as diff and expand, handles expressions containing bernoulli.

Find the first and second derivatives of the Bernoulli polynomial:

syms n x
diff(bernoulli(n,x^2), x)
ans =
2*n*x*bernoulli(n - 1, x^2)
diff(bernoulli(n,x^2), x, x)
ans =
2*n*bernoulli(n - 1, x^2) +...
4*n*x^2*bernoulli(n - 2, x^2)*(n - 1)

Expand these expressions containing the Bernoulli polynomials:

expand(bernoulli(n, x + 3))
ans =
bernoulli(n, x) + (n*(x + 1)^n)/(x + 1) +...
(n*(x + 2)^n)/(x + 2) + (n*x^n)/x
expand(bernoulli(n, 3*x))
ans =
(3^n*bernoulli(n, x))/3 + (3^n*bernoulli(n, x + 1/3))/3 +...
(3^n*bernoulli(n, x + 2/3))/3

Input Arguments

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Index of the Bernoulli number or polynomial, specified as a nonnegative integer, symbolic nonnegative integer, variable, expression, function, vector, or matrix. If n is a vector or matrix, bernoulli returns Bernoulli numbers or polynomials for each element of n. If one input argument is a scalar and the other one is a vector or a matrix, bernoulli(n,x) expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.

Polynomial variable, specified as a symbolic variable, expression, function, vector, or matrix. If x is a vector or matrix, bernoulli returns Bernoulli numbers or polynomials for each element of x. When you use the bernoulli function to find Bernoulli polynomials, at least one argument must be a scalar or both arguments must be vectors or matrices of the same size. If one input argument is a scalar and the other one is a vector or a matrix, bernoulli(n,x) expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.

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Bernoulli Polynomials

The Bernoulli polynomials are defined as follows:

$\frac{t{e}^{xt}}{{e}^{t}-1}=\sum _{n=0}^{\infty }\mathrm{bernoulli}\left(n,x\right)\frac{{t}^{n}}{n!}\text{\hspace{0.17em}}$

Bernoulli Numbers

The Bernoulli numbers are defined as follows:

$\mathrm{bernoulli}\left(n\right)=\mathrm{bernoulli}\left(n,0\right)$