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diff --git a/gsl-1.9/doc/specfunc-gamma.texi b/gsl-1.9/doc/specfunc-gamma.texi new file mode 100644 index 0000000..f15e7ef --- /dev/null +++ b/gsl-1.9/doc/specfunc-gamma.texi @@ -0,0 +1,314 @@ +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 + + + + + |