The  functions described in this section are declared
in the header file @file{gsl_sf_gamma.h}.

@menu
* Gamma Functions::             
* Factorials::                  
* Pochhammer Symbol::           
* Incomplete Gamma Functions::  
* Beta Functions::              
* Incomplete Beta Function::    
@end menu

@node Gamma Functions
@subsection Gamma Functions
@cindex gamma functions

The Gamma function is defined by the following integral,
@tex
\beforedisplay
$$
\Gamma(x) = \int_0^{\infty} dt \, t^{x-1} \exp(-t)
$$
\afterdisplay
@end tex
@ifinfo

@example
\Gamma(x) = \int_0^\infty dt  t^@{x-1@} \exp(-t)
@end example

@end ifinfo
@noindent
It is related to the factorial function by @math{\Gamma(n)=(n-1)!}
for positive integer @math{n}.  Further information on the Gamma function
can be found in Abramowitz & Stegun, Chapter 6.  The functions
described in this section are declared in the header file
@file{gsl_sf_gamma.h}.

@deftypefun double gsl_sf_gamma (double @var{x})
@deftypefunx int gsl_sf_gamma_e (double @var{x}, gsl_sf_result * @var{result})
These routines compute the Gamma function @math{\Gamma(x)}, subject to @math{x}
not being a negative integer or zero.  The function is computed using the real
Lanczos method. The maximum value of @math{x} such that @math{\Gamma(x)} is not
considered an overflow is given by the macro @code{GSL_SF_GAMMA_XMAX}
and is 171.0.
@comment exceptions: GSL_EDOM, GSL_EOVRFLW, GSL_EROUND
@end deftypefun

@deftypefun double gsl_sf_lngamma (double @var{x})
@deftypefunx int gsl_sf_lngamma_e (double @var{x}, gsl_sf_result * @var{result})
@cindex logarithm of Gamma function
These routines compute the logarithm of the Gamma function,
@math{\log(\Gamma(x))}, subject to @math{x} not being a negative
integer or zero.  For @math{x<0} the real part of @math{\log(\Gamma(x))} is
returned, which is equivalent to @math{\log(|\Gamma(x)|)}.  The function
is computed using the real Lanczos method.
@comment exceptions: GSL_EDOM, GSL_EROUND
@end deftypefun

@deftypefun int gsl_sf_lngamma_sgn_e (double @var{x}, gsl_sf_result * @var{result_lg}, double * @var{sgn})
This routine computes the sign of the gamma function and the logarithm of
its magnitude, subject to @math{x} not being a negative integer or zero.  The
function is computed using the real Lanczos method.  The value of the
gamma function can be reconstructed using the relation @math{\Gamma(x) =
sgn * \exp(resultlg)}.
@comment exceptions: GSL_EDOM, GSL_EROUND
@end deftypefun

@deftypefun double gsl_sf_gammastar (double @var{x})
@deftypefunx int gsl_sf_gammastar_e (double @var{x}, gsl_sf_result * @var{result})
@cindex Regulated Gamma function
These routines compute the regulated Gamma Function @math{\Gamma^*(x)}
for @math{x > 0}. The regulated gamma function is given by,
@tex
\beforedisplay
$$
\eqalign{
\Gamma^*(x) &= \Gamma(x)/(\sqrt{2\pi} x^{(x-1/2)} \exp(-x))\cr
            &= \left(1 + {1 \over 12x} + ...\right) \quad\hbox{for~} x\to \infty\cr
}
$$
\afterdisplay
@end tex
@ifinfo

@example
\Gamma^*(x) = \Gamma(x)/(\sqrt@{2\pi@} x^@{(x-1/2)@} \exp(-x))
            = (1 + (1/12x) + ...)  for x \to \infty
@end example
@end ifinfo
and is a useful suggestion of Temme.
@comment exceptions: GSL_EDOM
@end deftypefun

@deftypefun double gsl_sf_gammainv (double @var{x})
@deftypefunx int gsl_sf_gammainv_e (double @var{x}, gsl_sf_result * @var{result})
@cindex Reciprocal Gamma function
These routines compute the reciprocal of the gamma function,
@math{1/\Gamma(x)} using the real Lanczos method.
@comment exceptions: GSL_EUNDRFLW, GSL_EROUND
@end deftypefun

@deftypefun int gsl_sf_lngamma_complex_e (double @var{zr}, double @var{zi}, gsl_sf_result * @var{lnr}, gsl_sf_result * @var{arg})
@cindex Complex Gamma function
This routine computes @math{\log(\Gamma(z))} for complex @math{z=z_r+i
z_i} and @math{z} not a negative integer or zero, using the complex Lanczos
method.  The returned parameters are @math{lnr = \log|\Gamma(z)|} and
@math{arg = \arg(\Gamma(z))} in @math{(-\pi,\pi]}.  Note that the phase
part (@var{arg}) is not well-determined when @math{|z|} is very large,
due to inevitable roundoff in restricting to @math{(-\pi,\pi]}.  This
will result in a @code{GSL_ELOSS} error when it occurs.  The absolute
value part (@var{lnr}), however, never suffers from loss of precision.
@comment exceptions: GSL_EDOM, GSL_ELOSS
@end deftypefun

@node Factorials
@subsection Factorials
@cindex factorial

Although factorials can be computed from the Gamma function, using
the relation @math{n! = \Gamma(n+1)} for non-negative integer @math{n},
it is usually more efficient to call the functions in this section,
particularly for small values of @math{n}, whose factorial values are
maintained in hardcoded tables.

@deftypefun double gsl_sf_fact (unsigned int @var{n})
@deftypefunx int gsl_sf_fact_e (unsigned int @var{n}, gsl_sf_result * @var{result})
@cindex factorial
These routines compute the factorial @math{n!}.  The factorial is
related to the Gamma function by @math{n! = \Gamma(n+1)}.
The maximum value of @math{n} such that @math{n!} is not
considered an overflow is given by the macro @code{GSL_SF_FACT_NMAX}
and is 170.
@comment exceptions: GSL_EDOM, GSL_OVRFLW
@end deftypefun

@deftypefun double gsl_sf_doublefact (unsigned int @var{n})
@deftypefunx int gsl_sf_doublefact_e (unsigned int @var{n}, gsl_sf_result * @var{result})
@cindex double factorial
These routines compute the double factorial @math{n!! = n(n-2)(n-4) \dots}. 
The maximum value of @math{n} such that @math{n!!} is not
considered an overflow is given by the macro @code{GSL_SF_DOUBLEFACT_NMAX}
and is 297.
@comment exceptions: GSL_EDOM, GSL_OVRFLW
@end deftypefun

@deftypefun double gsl_sf_lnfact (unsigned int @var{n})
@deftypefunx int gsl_sf_lnfact_e (unsigned int @var{n}, gsl_sf_result * @var{result})
@cindex logarithm of factorial
These routines compute the logarithm of the factorial of @var{n},
@math{\log(n!)}.  The algorithm is faster than computing
@math{\ln(\Gamma(n+1))} via @code{gsl_sf_lngamma} for @math{n < 170},
but defers for larger @var{n}.
@comment exceptions: none
@end deftypefun

@deftypefun double gsl_sf_lndoublefact (unsigned int @var{n})
@deftypefunx int gsl_sf_lndoublefact_e (unsigned int @var{n}, gsl_sf_result * @var{result})
@cindex logarithm of double factorial
These routines compute the logarithm of the double factorial of @var{n},
@math{\log(n!!)}.
@comment exceptions: none
@end deftypefun

@deftypefun double gsl_sf_choose (unsigned int @var{n}, unsigned int @var{m})
@deftypefunx int gsl_sf_choose_e (unsigned int @var{n}, unsigned int @var{m}, gsl_sf_result * @var{result})
@cindex combinatorial factor C(m,n)
These routines compute the combinatorial factor @code{n choose m}
@math{= n!/(m!(n-m)!)}
@comment exceptions: GSL_EDOM, GSL_EOVRFLW
@end deftypefun


@deftypefun double gsl_sf_lnchoose (unsigned int @var{n}, unsigned int @var{m})
@deftypefunx int gsl_sf_lnchoose_e (unsigned int @var{n}, unsigned int @var{m}, gsl_sf_result * @var{result})
@cindex logarithm of combinatorial factor C(m,n)
These routines compute the logarithm of @code{n choose m}.  This is
equivalent to the sum @math{\log(n!) - \log(m!) - \log((n-m)!)}.
@comment exceptions: GSL_EDOM 
@end deftypefun

@deftypefun double gsl_sf_taylorcoeff (int @var{n}, double @var{x})
@deftypefunx int gsl_sf_taylorcoeff_e (int @var{n}, double @var{x}, gsl_sf_result * @var{result})
@cindex Taylor coefficients, computation of
These routines compute the Taylor coefficient @math{x^n / n!} for 
@c{$x \ge 0$}
@math{x >= 0}, 
@c{$n \ge 0$}
@math{n >= 0}.
@comment exceptions: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
@end deftypefun

@node Pochhammer Symbol
@subsection Pochhammer Symbol

@deftypefun double gsl_sf_poch (double @var{a}, double @var{x})
@deftypefunx int gsl_sf_poch_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result})
@cindex Pochhammer symbol
@cindex Apell symbol, see Pochammer symbol
These routines compute the Pochhammer symbol @math{(a)_x = \Gamma(a +
x)/\Gamma(a)}, subject to @math{a} and @math{a+x} not being negative
integers or zero. The Pochhammer symbol is also known as the Apell symbol and
sometimes written as @math{(a,x)}.
@comment exceptions:  GSL_EDOM, GSL_EOVRFLW
@end deftypefun


@deftypefun double gsl_sf_lnpoch (double @var{a}, double @var{x})
@deftypefunx int gsl_sf_lnpoch_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result})
@cindex logarithm of Pochhammer symbol
These routines compute the logarithm of the Pochhammer symbol,
@math{\log((a)_x) = \log(\Gamma(a + x)/\Gamma(a))} for @math{a > 0},
@math{a+x > 0}.
@comment exceptions:  GSL_EDOM
@end deftypefun

@deftypefun int gsl_sf_lnpoch_sgn_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result}, double * @var{sgn})
These routines compute the sign of the Pochhammer symbol and the
logarithm of its magnitude.  The computed parameters are @math{result =
\log(|(a)_x|)} and @math{sgn = \sgn((a)_x)} where @math{(a)_x =
\Gamma(a + x)/\Gamma(a)}, subject to @math{a}, @math{a+x} not being
negative integers or zero.
@comment exceptions:  GSL_EDOM
@end deftypefun

@deftypefun double gsl_sf_pochrel (double @var{a}, double @var{x})
@deftypefunx int gsl_sf_pochrel_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result})
@cindex relative Pochhammer symbol
These routines compute the relative Pochhammer symbol @math{((a)_x -
1)/x} where @math{(a)_x = \Gamma(a + x)/\Gamma(a)}.
@comment exceptions:  GSL_EDOM
@end deftypefun


@node Incomplete Gamma Functions
@subsection Incomplete Gamma Functions

@deftypefun double gsl_sf_gamma_inc (double @var{a}, double @var{x})
@deftypefunx int gsl_sf_gamma_inc_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result})
@cindex non-normalized incomplete Gamma function
@cindex unnormalized incomplete Gamma function
These functions compute the unnormalized incomplete Gamma Function
@c{$\Gamma(a,x) = \int_x^\infty dt\, t^{(a-1)} \exp(-t)$}
@math{\Gamma(a,x) = \int_x^\infty dt t^@{a-1@} \exp(-t)}
for @math{a} real and @c{$x \ge 0$}
@math{x >= 0}.
@comment exceptions: GSL_EDOM
@end deftypefun

@deftypefun double gsl_sf_gamma_inc_Q (double @var{a}, double @var{x})
@deftypefunx int gsl_sf_gamma_inc_Q_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result})
@cindex incomplete Gamma function
These routines compute the normalized incomplete Gamma Function
@c{$Q(a,x) = 1/\Gamma(a) \int_x^\infty dt\, t^{(a-1)} \exp(-t)$}
@math{Q(a,x) = 1/\Gamma(a) \int_x^\infty dt t^@{a-1@} \exp(-t)}
for @math{a > 0}, @c{$x \ge 0$}
@math{x >= 0}.
@comment exceptions: GSL_EDOM
@end deftypefun

@deftypefun double gsl_sf_gamma_inc_P (double @var{a}, double @var{x})
@deftypefunx int gsl_sf_gamma_inc_P_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result})
@cindex complementary incomplete Gamma function
These routines compute the complementary normalized incomplete Gamma Function
@c{$P(a,x) = 1 - Q(a,x) = 1/\Gamma(a) \int_0^x dt\, t^{(a-1)} \exp(-t)$}
@math{P(a,x) = 1 - Q(a,x) = 1/\Gamma(a) \int_0^x dt t^@{a-1@} \exp(-t)}
for @math{a > 0}, @c{$x \ge 0$}
@math{x >= 0}. 

Note that Abramowitz & Stegun call @math{P(a,x)} the incomplete gamma
function (section 6.5).
@comment exceptions: GSL_EDOM
@end deftypefun

@node Beta Functions
@subsection Beta Functions

@deftypefun double gsl_sf_beta (double @var{a}, double @var{b})
@deftypefunx int gsl_sf_beta_e (double @var{a}, double @var{b}, gsl_sf_result * @var{result})
@cindex Beta function
These routines compute the Beta Function, @math{B(a,b) =
\Gamma(a)\Gamma(b)/\Gamma(a+b)} subject to @math{a} and @math{b} not
being negative integers.
@comment exceptions: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
@end deftypefun

@deftypefun double gsl_sf_lnbeta (double @var{a}, double @var{b})
@deftypefunx int gsl_sf_lnbeta_e (double @var{a}, double @var{b}, gsl_sf_result * @var{result})
@cindex logarithm of Beta function
These routines compute the logarithm of the Beta Function, @math{\log(B(a,b))}
subject to @math{a} and @math{b} not
being negative integers.
@comment exceptions: GSL_EDOM
@end deftypefun

@node Incomplete Beta Function
@subsection Incomplete Beta Function

@deftypefun double gsl_sf_beta_inc (double @var{a}, double @var{b}, double @var{x})
@deftypefunx int gsl_sf_beta_inc_e (double @var{a}, double @var{b}, double @var{x}, gsl_sf_result * @var{result})
@cindex incomplete Beta function, normalized
@cindex normalized incomplete Beta function
@cindex Beta function, incomplete normalized 
These routines compute the normalized incomplete Beta function
@math{I_x(a,b)=B_x(a,b)/B(a,b)} where @c{$B_x(a,b) = \int_0^x t^{a-1} (1-t)^{b-1} dt$}
@math{B_x(a,b) = \int_0^x t^@{a-1@} (1-t)^@{b-1@} dt}
for @math{a > 0}, @math{b > 0}, and @c{$0 \le x \le 1$}
@math{0 <= x <= 1}. 
@end deftypefun