Special mathematical functions related to the beta and gamma functions.

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

`a, b` |
non-negative numeric vectors. |

`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) = 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.

`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.

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.

`Arithmetic`

for simple, `sqrt`

for
miscellaneous mathematical functions and `Bessel`

for the
real Bessel functions.

For the incomplete gamma function see `pgamma`

.

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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