| Special {base} | R Documentation |
Special mathematical functions related to the beta and gamma functions.
beta(a, b) lbeta(a, b) gamma(x) lgamma(x) psigamma(x, deriv = 0) digamma(x) trigamma(x) choose(n, k) lchoose(n, k) factorial(x) lfactorial(x)
a, b, x, n |
numeric vectors. |
k, deriv |
integer vectors. |
The functions beta and lbeta return the beta function
and the natural logarithm of the beta function,
B(a,b) = (Gamma(a)Gamma(b))/(Gamma(a+b)).
The formal definition is
integral_0^1 t^(a-1) (1-t)^(b-1) dt
(Abramowitz and Stegun (6.2.1), page 258).
The functions gamma and lgamma return the gamma function
Γ(x) and the natural logarithm of the absolute value of the
gamma function. The gamma function is defined by
(Abramowitz and Stegun (6.1.1), page 255)
integral_0^Inf t^(a-1) exp(-t) dt
factorial(x) is x! and identical to
gamma(x+1) and lfactorial is lgamma(x+1).
The functions digamma and trigamma return the first and second
derivatives of the logarithm of the gamma function.
psigamma(x, deriv) (deriv >= 0) is more generally
computing the deriv-th derivative of psi(x).
digamma(x) = psi(x) = d/dx {ln Gamma(x)} = Gamma'(x) / Gamma(x)
The functions choose and lchoose return binomial
coefficients and their logarithms. Note that choose(n,k) is
defined for all real numbers n and integer k. For k >= 1 as n(n-1)...(n-k+1) / k!,
as 1 for k = 0 and as 0 for negative k.
choose(*,k) uses direct arithmetic (instead of
[l]gamma calls) for small k, for speed and accuracy
reasons. Note the function combn (package
utils) for enumeration of all possible combinations.
The gamma, lgamma, digamma and trigamma
functions are generic: methods can be defined for them individually or
via the Math group generic.
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
The New S Language.
Wadsworth & Brooks/Cole. (for gamma and lgamma.)
Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover. Chapter 6: Gamma and Related Functions.
Arithmetic for simple, sqrt for
miscellaneous mathematical functions and Bessel for the
real Bessel functions.
For the incomplete gamma function see pgamma.
choose(5, 2)
for (n in 0:10) print(choose(n, k = 0:n))
factorial(100)
lfactorial(10000)
## gamma has 1st order poles at 0, -1, -2, ...
## this will generate loss of precision warnings, so turn off
op <- options("warn")
options(warn = -1)
x <- sort(c(seq(-3,4, length=201), outer(0:-3, (-1:1)*1e-6, "+")))
plot(x, gamma(x), ylim=c(-20,20), col="red", type="l", lwd=2,
main=expression(Gamma(x)))
abline(h=0, v=-3:0, lty=3, col="midnightblue")
options(op)
x <- seq(.1, 4, length = 201); dx <- diff(x)[1]
par(mfrow = c(2, 3))
for (ch in c("", "l","di","tri","tetra","penta")) {
is.deriv <- nchar(ch) >= 2
nm <- paste(ch, "gamma", sep = "")
if (is.deriv) {
dy <- diff(y) / dx # finite difference
der <- which(ch == c("di","tri","tetra","penta")) - 1
nm2 <- paste("psigamma(*, deriv = ", der,")",sep='')
nm <- if(der >= 2) nm2 else paste(nm, nm2, sep = " ==\n")
y <- psigamma(x, deriv=der)
} else {
y <- get(nm)(x)
}
plot(x, y, type = "l", main = nm, col = "red")
abline(h = 0, col = "lightgray")
if (is.deriv) lines(x[-1], dy, col = "blue", lty = 2)
}
par(mfrow = c(1, 1))
## "Extended" Pascal triangle:
fN <- function(n) formatC(n, wid=2)
for (n in -4:10) cat(fN(n),":", fN(choose(n, k= -2:max(3,n+2))), "\n")
## R code version of choose() [simplistic; warning for k < 0]:
mychoose <- function(r,k)
ifelse(k <= 0, (k==0),
sapply(k, function(k) prod(r:(r-k+1))) / factorial(k))
k <- -1:6
cbind(k=k, choose(1/2, k), mychoose(1/2, k))
## Binomial theorem for n=1/2 ;
## sqrt(1+x) = (1+x)^(1/2) = sum_{k=0}^Inf choose(1/2, k) * x^k :
k <- 0:10 # 10 is sufficient for ~ 9 digit precision:
sqrt(1.25)
sum(choose(1/2, k)* .25^k)