@@ -52,23 +52,23 @@ lfactorial(x)
5252 \deqn {B(a ,b ) = \frac {\Gamma(a )\Gamma(b )}{\Gamma(a + b )}. }{B(a ,b ) = \Gamma(a )\Gamma(b )/ \Gamma(a + b ). }
5353 The formal definition is
5454 \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 }
55- \bibcitep {| R : Abramowitz + Stegun : 1972 | section 6.2.1 \\\\\\ , page 258 }.
55+ \bibcitep {| R : Abramowitz + Stegun : 1972 | section 6.2.1 \, page 258 }.
5656 Note that it is only
5757 defined in \R for non - negative \code {a } and \code {b }, and is infinite
5858 if either is zero.
5959
6060 The functions \code {gamma } and \code {lgamma } return the gamma function
6161 \eqn {\Gamma(x )} and the natural logarithm of \emph {the absolute value of } the
6262 gamma function . The gamma function is defined by
63- \bibcitep {| R : Abramowitz + Stegun : 1972 | section 6.1.1 \\\\\\ , page 255 }
63+ \bibcitep {| R : Abramowitz + Stegun : 1972 | section 6.1.1 \, page 255 }
6464 \deqn {\Gamma(x ) = \int_0 ^ \infty t ^ {x - 1 } e ^ {- t } dt }{\Gamma(x ) = integral_0 ^ Inf t ^ (x - 1 ) exp(- t ) dt }
6565 for all \eqn {x > 0 }, from which the recursions \eqn {\Gamma(x + 1 ) =
6666 x \Gamma(x )} and then \eqn {\Gamma(x + n ) = (x + n - 1 )(x + n - 2 )\cdots x \Gamma(x )}
6767 for all non - negative integers \eqn {n }. Solving for \eqn {\Gamma(x )} and
6868 analytic continuation leads to the expression for non - integer negative real numbers ,
6969 \deqn {\Gamma(x ) = \frac {\Gamma(x + n )}{(x + n - 1 ) \cdots (x + 1 )x }, \ n \in \mathbb {Z }^ {+ }, - n < x < 0 ,%
7070 }{\Gamma(x ) = \Gamma(x + n )/ ((x + n - 1 ) ... (x + 1 )x ), n in N , - n < x < 0 ,}
71- see \bibcitet {| R : Abramowitz + Stegun : 1972 | 6.1.16 or 6.1.22 \\\\\\ , page 256 }.
71+ see \bibcitet {| R : Abramowitz + Stegun : 1972 | 6.1.16 or 6.1.22 \, page 256 }.
7272 %
7373 The gamma function is not defined for zero and negative integers (when
7474 \code {NaN } is returned ). There will be a warning on possible loss of
@@ -87,7 +87,7 @@ lfactorial(x)
8787 \frac {\Gamma ' (x)}{\G amma(x)}}{digamma(x) = \p si(x) = d/dx{ln \G amma(x)} = \G amma' (x ) / \Gamma(x )}
8888 \eqn {\psi } and its derivatives , the \code {psigamma()} functions , are
8989 often called the \sQuote {\I {polygamma }} functions , e.g. \sspace {}in
90- \bibcitet {| R : Abramowitz + Stegun : 1972 | section 6.4.1 \\\\\\ , page 260 }; and higher
90+ \bibcitet {| R : Abramowitz + Stegun : 1972 | section 6.4.1 \, page 260 }; and higher
9191 derivatives (\code {deriv = 2 : 4 }) have occasionally been called
9292 \sQuote {\I {tetragamma }}, \sQuote {\I {pentagamma }}, and \sQuote {\I {hexagamma }}.
9393
@@ -120,10 +120,10 @@ lfactorial(x)
120120}
121121\references {
122122 \bibinfo {R : Abramowitz + Stegun : 1972 }{footer }{
123- \\\\\\ url {https : // en.wikipedia.org / wiki / Abramowitz_and_Stegun } provides
124- links to the full text which is in public domain. \\\\\\ cr
123+ \url {https : // en.wikipedia.org / wiki / Abramowitz_and_Stegun } provides
124+ links to the full text which is in public domain. \cr
125125 Chapter 6 : Gamma and Related Functions. }
126- \bibinfo {R : Becker + Chambers + Wilks : 1988 }{footer }{(For \\\\\\ code {gamma } and \\\\\ \code {lgamma }. )}
126+ \bibinfo {R : Becker + Chambers + Wilks : 1988 }{footer }{(For \code {gamma } and \code {lgamma }. )}
127127 \bibshow {* , R : Becker + Chambers + Wilks : 1988 }
128128}
129129\seealso {
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