Special: Special Functions of Mathematics

Description Usage Arguments Details Source References See Also Examples

Description

Special mathematical functions related to the beta and gamma functions.

Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```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

 `a, b` non-negative numeric vectors. `x, n` numeric vectors. `k, deriv` integer vectors.

Details

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

B(a,b) = Γ(a)Γ(b)/Γ(a+b).

The formal definition is

integral_0^1 t^(a-1) (1-t)^(b-1) dt

(Abramowitz and Stegun section 6.2.1, page 258). Note that it is only defined in R for non-negative `a` and `b`, and is infinite if either is zero.

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 section 6.1.1, page 255)

Γ(x) = integral_0^Inf t^(x-1) exp(-t) dt

for all real `x` except zero and negative integers (when `NaN` is returned). There will be a warning on possible loss of precision for values which are too close (within about 1e-8)) to a negative integer less than -10.

`factorial(x)` (x! for non-negative integer `x`) is defined to be `gamma(x+1)` and `lfactorial` to be `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`) computes the `deriv`-th derivative of ψ(x).

digamma(x) = ψ(x) = d/dx{ln Γ(x)} = Γ'(x) / Γ(x)

ψ and its derivatives, the `psigamma()` functions, are often called the ‘polygamma’ functions, e.g. in Abramowitz and Stegun (section 6.4.1, page 260); and higher derivatives (`deriv = 2:4`) have occasionally been called ‘tetragamma’, ‘pentagamma’, and ‘hexagamma’.

The functions `choose` and `lchoose` return binomial coefficients and the logarithms of their absolute values. Note that `choose(n, k)` is defined for all real numbers n and integer k. For k ≥ 1 it is defined as n(n-1)…(n-k+1) / k!, as 1 for k = 0 and as 0 for negative k. Non-integer values of `k` are rounded to an integer, with a warning.
`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 internal generic primitive functions: methods can be defined for them individually or via the `Math` group generic.

Source

`gamma`, `lgamma`, `beta` and `lbeta` are based on C translations of Fortran subroutines by W. Fullerton of Los Alamos Scientific Laboratory (now available as part of SLATEC).

`digamma`, `trigamma` and `psigamma` are based on

Amos, D. E. (1983). A portable Fortran subroutine for derivatives of the psi function, Algorithm 610, ACM Transactions on Mathematical Software 9(4), 494–502.

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. https://en.wikipedia.org/wiki/Abramowitz_and_Stegun provides links to the full text which is in public domain.
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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57``` ```require(graphics) 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.out = 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(0.1, 4, length.out = 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 <- paste0(ch, "gamma") if (is.deriv) { dy <- diff(y) / dx # finite difference der <- which(ch == c("di","tri","tetra","penta")) - 1 nm2 <- paste0("psigamma(*, deriv = ", der,")") 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, width=2) for (n in -4:10) { cat(fN(n),":", fN(choose(n, k = -2:max(3, n+2)))) cat("\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) ```

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