Bernoulli Numbers and Polynomials
The Bernoulli numbers are a sequence of rational numbers that play an important role for the series expansion of hyperbolic functions, in the Euler-MacLaurin formula, or for certain values of Riemann's function at negative integers.
bernoulli(n, x)
n |
the index, a whole number greater or equal to 0. |
x |
real number or vector of real numbers; if missing, the Bernoulli numbers will be given, otherwise the polynomial. |
The calculation of the Bernoulli numbers uses the values of the zeta function
at negative integers, i.e. B_n = -n \, zeta(1-n). Bernoulli numbers
B_n for odd n are 0 except B_1 which is set to -0.5 on
purpose.
The Bernoulli polynomials can be directly defined as
B_n(x) = ∑_{k=0}^n {n \choose k} b_{n-k}\, x^k
and it is immediately clear that the Bernoulli numbers are then given as B_n = B_n(0).
Returns the first n+1 Bernoulli numbers, if x is missing, or
the value of the Bernoulli polynomial at point(s) x.
The definition uses B_1 = -1/2 in accordance with the definition of
the Bernoulli polynomials.
See the entry on Bernoulli numbers in the Wikipedia.
bernoulli(10)
# 1.00000000 -0.50000000 0.16666667 0.00000000 -0.03333333
# 0.00000000 0.02380952 0.00000000 -0.03333333 0.00000000 0.07575758
#
## Not run:
x1 <- linspace(0.3, 0.7, 2)
y1 <- bernoulli(1, x1)
plot(x1, y1, type='l', col='red', lwd=2,
xlim=c(0.0, 1.0), ylim=c(-0.2, 0.2),
xlab="", ylab="", main="Bernoulli Polynomials")
grid()
xs <- linspace(0, 1, 51)
lines(xs, bernoulli(2, xs), col="green", lwd=2)
lines(xs, bernoulli(3, xs), col="blue", lwd=2)
lines(xs, bernoulli(4, xs), col="cyan", lwd=2)
lines(xs, bernoulli(5, xs), col="brown", lwd=2)
lines(xs, bernoulli(6, xs), col="magenta", lwd=2)
legend(0.75, 0.2, c("B_1", "B_2", "B_3", "B_4", "B_5", "B_6"),
col=c("red", "green", "blue", "cyan", "brown", "magenta"),
lty=1, lwd=2)
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.