Special Functions of Mathematics

Description

The functions beta and lbeta return the beta function and the natural logarithm of the beta function,

B(a,b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}.

The functions gamma and lgamma return the gamma function \Gamma(x) and the natural logarithm of the absolute value of the gamma function.

The functions digamma, trigamma, tetragamma and pentagamma return the first, second, third and fourth derivatives of the logarithm of the gamma function.

\code{digamma(x)} = \psi(x) = \frac{d}{dx}\ln\Gamma(x) = \frac{\Gamma'(x)}{\Gamma(x)}

The functions choose and lchoose return binomial coefficients and their logarithms.

Usage

beta(a, b)
lbeta(a, b)
gamma(x)
lgamma(x)
digamma(x)
trigamma(x)
tetragamma(x)
pentagamma(x)
choose(n,k)
lchoose(n,k)

References

Abramowitz, M. and Stegun, I. A. (1972).
Handbook of Mathematical Functions, Dover, New York;
Chapter 6: Gamma and Related Functions.

See Also

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

Examples

choose(5, 2)
for (n in 0:10) print(choose(n, k = 0:n))

curve(gamma(x),-3,4, n=1001, ylim=c(-10,100),
      col="red", lwd=2, main="gamma(x)")
abline(h=0,v=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
  if (is.deriv) dy <- diff(y) / dx
  nm <- paste(ch, "gamma", sep = "")
  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(2, 2))