Special Functions of Mathematics

Description

Special mathematical functions related to the beta and gamma functions.

Usage

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)

Arguments

Details

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.

The gamma, lgamma, digamma and trigamma functions are generic: methods can be defined for them individually or via the Math group generic.

References

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.

See Also

Arithmetic for simple, sqrt for miscellaneous mathematical functions and Bessel for the real Bessel functions.

For the incomplete gamma function see pgamma.

Examples

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, ...
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")

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)
}

## "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)