Special {base} | R Documentation |
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 functions gamma
and lgamma
return the gamma function
Γ(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.
digamma(x)
= psi(x) = d/dx {ln Gamma(x)} = Gamma'(x) / Gamma(x)
The functions choose
and lchoose
return binomial
coefficients and their logarithms.
beta(a, b) lbeta(a, b) gamma(x) lgamma(x) digamma(x) trigamma(x) tetragamma(x) pentagamma(x) choose(n,k) lchoose(n,k)
Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover. Chapter 6: Gamma and Related Functions.
Arithmetic
for simple, Math
for
miscellaneous mathematical functions and Bessel
for the
real Bessel functions.
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))