1 /** @file inifcns_gamma.cpp
3 * Implementation of Gamma-function, Polygamma-functions, and some related
7 * GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
9 * This program is free software; you can redistribute it and/or modify
10 * it under the terms of the GNU General Public License as published by
11 * the Free Software Foundation; either version 2 of the License, or
12 * (at your option) any later version.
14 * This program is distributed in the hope that it will be useful,
15 * but WITHOUT ANY WARRANTY; without even the implied warranty of
16 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
17 * GNU General Public License for more details.
19 * You should have received a copy of the GNU General Public License
20 * along with this program; if not, write to the Free Software
21 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
40 /** Evaluation of gamma(x). Knows about integer arguments, half-integer
41 * arguments and that's it. Somebody ought to provide some good numerical
42 * evaluation some day...
44 * @exception fail_numeric("complex_infinity") or something similar... */
45 static ex gamma_eval(ex const & x)
47 if (x.info(info_flags::numeric)) {
48 // trap integer arguments:
49 if (x.info(info_flags::integer)) {
50 // gamma(n+1) -> n! for postitive n
51 if (x.info(info_flags::posint)) {
52 return factorial(ex_to_numeric(x).sub(numONE()));
54 return numZERO(); // Infinity. Throw? What?
57 // trap half integer arguments:
58 if ((x*2).info(info_flags::integer)) {
59 // trap positive x=(n+1/2)
60 // gamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n)
61 if ((x*2).info(info_flags::posint)) {
62 numeric n = ex_to_numeric(x).sub(numHALF());
63 numeric coefficient = doublefactorial(n.mul(numTWO()).sub(numONE()));
64 coefficient = coefficient.div(numTWO().power(n));
65 return coefficient * pow(Pi,numHALF());
67 // trap negative x=(-n+1/2)
68 // gamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1))
69 numeric n = abs(ex_to_numeric(x).sub(numHALF()));
70 numeric coefficient = numeric(-2).power(n);
71 coefficient = coefficient.div(doublefactorial(n.mul(numTWO()).sub(numONE())));;
72 return coefficient*sqrt(Pi);
76 return gamma(x).hold();
79 static ex gamma_evalf(ex const & x)
83 END_TYPECHECK(gamma(x))
85 return gamma(ex_to_numeric(x));
88 static ex gamma_diff(ex const & x, unsigned diff_param)
90 GINAC_ASSERT(diff_param==0);
92 return psi(x)*gamma(x); // diff(log(gamma(x)),x)==psi(x)
95 static ex gamma_series(ex const & x, symbol const & s, ex const & point, int order)
97 // FIXME: Only handle one special case for now...
98 if (x.is_equal(s) && point.is_zero()) {
99 ex e = 1 / s - EulerGamma + s * (pow(Pi, 2) / 12 + pow(EulerGamma, 2) / 2) + Order(pow(s, 2));
100 return e.series(s, point, order);
102 throw(std::logic_error("don't know the series expansion of this particular gamma function"));
105 REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series);
108 // Psi-function (aka polygamma-function)
111 /** Evaluation of polygamma-function psi(x).
112 * Somebody ought to provide some good numerical evaluation some day... */
113 static ex psi1_eval(ex const & x)
115 if (x.info(info_flags::numeric)) {
118 return psi(x).hold();
121 static ex psi1_evalf(ex const & x)
125 END_TYPECHECK(psi(x))
127 return psi(ex_to_numeric(x));
130 static ex psi1_diff(ex const & x, unsigned diff_param)
132 GINAC_ASSERT(diff_param==0);
134 return psi(exONE(), x);
137 static ex psi1_series(ex const & x, symbol const & s, ex const & point, int order)
139 throw(std::logic_error("Nobody told me how to series expand the psi function. :-("));
142 unsigned function_index_psi1 = function::register_new("psi", psi1_eval, psi1_evalf, psi1_diff, psi1_series);
145 // Psi-functions (aka polygamma-functions) psi(0,x)==psi(x)
148 /** Evaluation of polygamma-function psi(n,x).
149 * Somebody ought to provide some good numerical evaluation some day... */
150 static ex psi2_eval(ex const & n, ex const & x)
152 // psi(0,x) -> psi(x)
154 return psi(x).hold();
155 if (n.info(info_flags::numeric) && x.info(info_flags::numeric)) {
158 return psi(n, x).hold();
161 static ex psi2_evalf(ex const & n, ex const & x)
166 END_TYPECHECK(psi(n,x))
168 return psi(ex_to_numeric(n), ex_to_numeric(x));
171 static ex psi2_diff(ex const & n, ex const & x, unsigned diff_param)
173 GINAC_ASSERT(diff_param<2);
177 throw(std::logic_error("cannot diff psi(n,x) with respect to n"));
183 static ex psi2_series(ex const & n, ex const & x, symbol const & s, ex const & point, int order)
185 throw(std::logic_error("Nobody told me how to series expand the psi functions. :-("));
188 unsigned function_index_psi2 = function::register_new("psi", psi2_eval, psi2_evalf, psi2_diff, psi2_series);