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The functions \code{beta} and \code{lbeta} return the beta function
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The functions \code{beta} and \code{lbeta} return the beta function
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and the natural logarithm of the beta function,
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and the natural logarithm of the beta function,
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\deqn{B(a,b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}.}{B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b).}
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\deqn{B(a,b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}.}{B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b).}
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The formal definition is
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The formal definition is
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\deqn{B(a, b) = \int_0^1 t^{a-1} (1-t)^{b-1} dt}{integral_0^1 t^(a-1) (1-t)^(b-1) dt}
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\deqn{B(a, b) = \int_0^1 t^{a-1} (1-t)^{b-1} dt}{integral_0^1 t^(a-1) (1-t)^(b-1) dt}
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\bibcitep{|R:Abramowitz+Stegun:1972|section 6.2.1\, page 258}.
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\bibcitep{|R:Abramowitz+Stegun:1972|page 258 in section 6.2.1}.
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Note that it is only
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Note that it is only
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defined in \R for non-negative \code{a} and \code{b}, and is infinite
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defined in \R for non-negative \code{a} and \code{b}, and is infinite
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if either is zero.
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if either is zero.
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The functions \code{gamma} and \code{lgamma} return the gamma function
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The functions \code{gamma} and \code{lgamma} return the gamma function
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\eqn{\Gamma(x)} and the natural logarithm of \emph{the absolute value of} the
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\eqn{\Gamma(x)} and the natural logarithm of \emph{the absolute value of} the
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gamma function. The gamma function is defined by
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gamma function. The gamma function is defined by
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\bibcitep{|R:Abramowitz+Stegun:1972|section 6.1.1\, page 255}
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\bibcitep{|R:Abramowitz+Stegun:1972|page 255 in section 6.1.1}
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\deqn{\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} dt}{\Gamma(x) = integral_0^Inf t^(x-1) exp(-t) dt}
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\deqn{\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} dt}{\Gamma(x) = integral_0^Inf t^(x-1) exp(-t) dt}
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for all \eqn{x > 0}, from which the recursions \eqn{\Gamma(x+1) =
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for all \eqn{x > 0}, from which the recursions \eqn{\Gamma(x+1) =
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x\Gamma(x)} and then \eqn{\Gamma(x+n) = (x+n-1)(x+n-2)\cdots x \Gamma(x)}
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x\Gamma(x)} and then \eqn{\Gamma(x+n) = (x+n-1)(x+n-2)\cdots x \Gamma(x)}
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for all non-negative integers \eqn{n}. Solving for \eqn{\Gamma(x)} and
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for all non-negative integers \eqn{n}. Solving for \eqn{\Gamma(x)} and
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analytic continuation leads to the expression for non-integer negative real numbers,
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analytic continuation leads to the expression for non-integer negative real numbers,
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\deqn{\Gamma(x) = \frac{\Gamma(x + n)}{(x + n -1) \cdots (x + 1)x}, \ n \in \mathbb{Z}^{+}, -n < x < 0,%
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\deqn{\Gamma(x) = \frac{\Gamma(x + n)}{(x + n -1) \cdots (x + 1)x}, \ n \in \mathbb{Z}^{+}, -n < x < 0,%
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}{\Gamma(x) = \Gamma(x + n)/((x + n -1) ... (x + 1)x), n in N, -n < x < 0,}
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}{\Gamma(x) = \Gamma(x + n)/((x + n -1) ... (x + 1)x), n in N, -n < x < 0,}
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see \bibcitet{|R:Abramowitz+Stegun:1972|6.1.16 or 6.1.22\, page 256}.
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see \bibcitet{|R:Abramowitz+Stegun:1972|equations 6.1.16 or 6.1.22 on page 256}.
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%
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%
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The gamma function is not defined for zero and negative integers (when
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The gamma function is not defined for zero and negative integers (when
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\code{NaN} is returned). There will be a warning on possible loss of
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\code{NaN} is returned). There will be a warning on possible loss of
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precision for values which are too close (within about
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precision for values which are too close (within about
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\eqn{10^{-8}}{1e-8}) to a negative integer less than \samp{-10}.
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\eqn{10^{-8}}{1e-8}) to a negative integer less than \samp{-10}.
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\code{deriv}-th derivative of \eqn{\psi(x)}.
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\code{deriv}-th derivative of \eqn{\psi(x)}.
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\deqn{\code{digamma(x)} = \psi(x) = \frac{d}{dx}\ln\Gamma(x) =
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\deqn{\code{digamma(x)} = \psi(x) = \frac{d}{dx}\ln\Gamma(x) =
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\frac{\Gamma'(x)}{\Gamma(x)}}{digamma(x) = \psi(x) = d/dx{ln \Gamma(x)} = \Gamma'(x) / \Gamma(x)}
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\frac{\Gamma'(x)}{\Gamma(x)}}{digamma(x) = \psi(x) = d/dx{ln \Gamma(x)} = \Gamma'(x) / \Gamma(x)}
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\eqn{\psi} and its derivatives, the \code{psigamma()} functions, are
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\eqn{\psi} and its derivatives, the \code{psigamma()} functions, are
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often called the \sQuote{\I{polygamma}} functions, e.g.\sspace{}in
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often called the \sQuote{\I{polygamma}} functions, e.g.\sspace{}in
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\bibcitet{|R:Abramowitz+Stegun:1972|section 6.4.1\, page 260}; and higher
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\bibcitet{|R:Abramowitz+Stegun:1972|page 260 in section 6.4.1}; and higher
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derivatives (\code{deriv = 2:4}) have occasionally been called
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derivatives (\code{deriv = 2:4}) have occasionally been called
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\sQuote{\I{tetragamma}}, \sQuote{\I{pentagamma}}, and \sQuote{\I{hexagamma}}.
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\sQuote{\I{tetragamma}}, \sQuote{\I{pentagamma}}, and \sQuote{\I{hexagamma}}.
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The functions \code{choose} and \code{lchoose} return binomial
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The functions \code{choose} and \code{lchoose} return binomial
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coefficients and the logarithms of their absolute values. Note that
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coefficients and the logarithms of their absolute values. Note that
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