/** @file inifcns_gamma.cpp * * Implementation of Gamma function and some related stuff. * * GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation; either version 2 of the License, or * (at your option) any later version. * * This program is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ #include #include #include "ginac.h" ////////// // gamma function ////////// /** Evaluation of gamma(x). Knows about integer arguments, half-integer * arguments and that's it. Somebody ought to provide some good numerical * evaluation some day... * * @exception fail_numeric("complex_infinity") or something similar... */ ex gamma_eval(ex const & x) { if ( x.info(info_flags::numeric) ) { // trap integer arguments: if ( x.info(info_flags::integer) ) { // gamma(n+1) -> n! for postitive n if ( x.info(info_flags::posint) ) { return factorial(ex_to_numeric(x).sub(numONE())); } else { return numZERO(); // Infinity. Throw? What? } } // trap half integer arguments: if ( (x*2).info(info_flags::integer) ) { // trap positive x=(n+1/2) // gamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n) if ( (x*2).info(info_flags::posint) ) { numeric n = ex_to_numeric(x).sub(numHALF()); numeric coefficient = doublefactorial(n.mul(numTWO()).sub(numONE())); coefficient = coefficient.div(numTWO().power(n)); return mul(coefficient,power(Pi,numHALF())); } else { // trap negative x=(-n+1/2) // gamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1)) numeric n = abs(ex_to_numeric(x).sub(numHALF())); numeric coefficient = numeric(-2).power(n); coefficient = coefficient.div(doublefactorial(n.mul(numTWO()).sub(numONE())));; return mul(coefficient,power(Pi,numHALF())); } } } return gamma(x).hold(); } ex gamma_evalf(ex const & x) { BEGIN_TYPECHECK TYPECHECK(x,numeric) END_TYPECHECK(gamma(x)) return gamma(ex_to_numeric(x)); } ex gamma_diff(ex const & x, unsigned diff_param) { ASSERT(diff_param==0); return power(x, -1); //!! } ex gamma_series(ex const & x, symbol const & s, ex const & point, int order) { //!! Only handle one special case for now... if (x.is_equal(s) && point.is_zero()) { ex e = 1 / s - EulerGamma + s * (power(Pi, 2) / 12 + power(EulerGamma, 2) / 2) + Order(power(s, 2)); return e.series(s, point, order); } else throw(std::logic_error("don't know the series expansion of this particular gamma function")); } REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series);