X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_gamma.cpp;h=f10e55e460758340ecb98f226cff43263b2c2762;hp=9c1da65157426b2cb960fd29f7f491c2012b57f6;hb=25b4d894b5b2f7f56fb49e62a7767ab969d5249f;hpb=6bff4f32e5cbe9f7e51837f392a2d0bf0d5b721d diff --git a/ginac/inifcns_gamma.cpp b/ginac/inifcns_gamma.cpp index 9c1da651..f10e55e4 100644 --- a/ginac/inifcns_gamma.cpp +++ b/ginac/inifcns_gamma.cpp @@ -4,7 +4,7 @@ * some related stuff. */ /* - * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2016 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 @@ -18,22 +18,23 @@ * * 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 + * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ -#include -#include - #include "inifcns.h" #include "constant.h" #include "pseries.h" #include "numeric.h" #include "power.h" #include "relational.h" +#include "operators.h" #include "symbol.h" #include "symmetry.h" #include "utils.h" +#include +#include + namespace GiNaC { ////////// @@ -53,8 +54,7 @@ static ex lgamma_evalf(const ex & x) /** Evaluation of lgamma(x), the natural logarithm of the Gamma function. - * Knows about integer arguments and that's it. Somebody ought to provide - * some good numerical evaluation some day... + * Handles integer arguments as a special case. * * @exception GiNaC::pole_error("lgamma_eval(): logarithmic pole",0) */ static ex lgamma_eval(const ex & x) @@ -64,11 +64,12 @@ static ex lgamma_eval(const ex & x) if (x.info(info_flags::integer)) { // lgamma(n) -> log((n-1)!) for postitive n if (x.info(info_flags::posint)) - return log(factorial(x + _ex_1())); + return log(factorial(x + _ex_1)); else throw (pole_error("lgamma_eval(): logarithmic pole",0)); } - // lgamma_evalf should be called here once it becomes available + if (!ex_to(x).is_rational()) + return lgamma(ex_to(x)); } return lgamma(x).hold(); @@ -97,7 +98,7 @@ static ex lgamma_series(const ex & arg, // from which follows // series(lgamma(x),x==-m,order) == // series(lgamma(x+m+1)-log(x)...-log(x+m)),x==-m,order); - const ex arg_pt = arg.subs(rel); + const ex arg_pt = arg.subs(rel, subs_options::no_pattern); if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive)) throw do_taylor(); // caught by function::series() // if we got here we have to care for a simple pole of tgamma(-m): @@ -105,7 +106,22 @@ static ex lgamma_series(const ex & arg, ex recur; for (numeric p = 0; p<=m; ++p) recur += log(arg+p); - return (lgamma(arg+m+_ex1())-recur).series(rel, order, options); + return (lgamma(arg+m+_ex1)-recur).series(rel, order, options); +} + + +static ex lgamma_conjugate(const ex & x) +{ + // conjugate(lgamma(x))==lgamma(conjugate(x)) unless on the branch cut + // which runs along the negative real axis. + if (x.info(info_flags::positive)) { + return lgamma(x); + } + if (is_exactly_a(x) && + !x.imag_part().is_zero()) { + return lgamma(x.conjugate()); + } + return conjugate_function(lgamma(x)).hold(); } @@ -113,6 +129,7 @@ REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval). evalf_func(lgamma_evalf). derivative_func(lgamma_deriv). series_func(lgamma_series). + conjugate_func(lgamma_conjugate). latex_name("\\log \\Gamma")); @@ -141,11 +158,11 @@ static ex tgamma_eval(const ex & x) { if (x.info(info_flags::numeric)) { // trap integer arguments: - const numeric two_x = _num2()*ex_to(x); + const numeric two_x = (*_num2_p)*ex_to(x); if (two_x.is_even()) { // tgamma(n) -> (n-1)! for postitive n if (two_x.is_positive()) { - return factorial(ex_to(x).sub(_num1())); + return factorial(ex_to(x).sub(*_num1_p)); } else { throw (pole_error("tgamma_eval(): simple pole",1)); } @@ -155,16 +172,17 @@ static ex tgamma_eval(const ex & x) // trap positive x==(n+1/2) // tgamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n) if (two_x.is_positive()) { - const numeric n = ex_to(x).sub(_num1_2()); - return (doublefactorial(n.mul(_num2()).sub(_num1())).div(pow(_num2(),n))) * pow(Pi,_ex1_2()); + const numeric n = ex_to(x).sub(*_num1_2_p); + return (doublefactorial(n.mul(*_num2_p).sub(*_num1_p)).div(pow(*_num2_p,n))) * sqrt(Pi); } else { // trap negative x==(-n+1/2) // tgamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1)) - const numeric n = abs(ex_to(x).sub(_num1_2())); - return (pow(_num_2(), n).div(doublefactorial(n.mul(_num2()).sub(_num1()))))*power(Pi,_ex1_2()); + const numeric n = abs(ex_to(x).sub(*_num1_2_p)); + return (pow(*_num_2_p, n).div(doublefactorial(n.mul(*_num2_p).sub(*_num1_p))))*sqrt(Pi); } } - // tgamma_evalf should be called here once it becomes available + if (!ex_to(x).is_rational()) + return tgamma(ex_to(x)); } return tgamma(x).hold(); @@ -192,16 +210,23 @@ static ex tgamma_series(const ex & arg, // tgamma(x) == tgamma(x+1) / x // from which follows // series(tgamma(x),x==-m,order) == - // series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x==-m,order+1); - const ex arg_pt = arg.subs(rel); + // series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x==-m,order); + const ex arg_pt = arg.subs(rel, subs_options::no_pattern); if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive)) throw do_taylor(); // caught by function::series() // if we got here we have to care for a simple pole at -m: const numeric m = -ex_to(arg_pt); - ex ser_denom = _ex1(); + ex ser_denom = _ex1; for (numeric p; p<=m; ++p) ser_denom *= arg+p; - return (tgamma(arg+m+_ex1())/ser_denom).series(rel, order+1, options); + return (tgamma(arg+m+_ex1)/ser_denom).series(rel, order, options); +} + + +static ex tgamma_conjugate(const ex & x) +{ + // conjugate(tgamma(x))==tgamma(conjugate(x)) + return tgamma(x.conjugate()); } @@ -209,6 +234,7 @@ REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval). evalf_func(tgamma_evalf). derivative_func(tgamma_deriv). series_func(tgamma_series). + conjugate_func(tgamma_conjugate). latex_name("\\Gamma")); @@ -220,7 +246,7 @@ static ex beta_evalf(const ex & x, const ex & y) { if (is_exactly_a(x) && is_exactly_a(y)) { try { - return tgamma(ex_to(x))*tgamma(ex_to(y))/tgamma(ex_to(x+y)); + return exp(lgamma(ex_to(x))+lgamma(ex_to(y))-lgamma(ex_to(x+y))); } catch (const dunno &e) { } } @@ -230,23 +256,27 @@ static ex beta_evalf(const ex & x, const ex & y) static ex beta_eval(const ex & x, const ex & y) { + if (x.is_equal(_ex1)) + return 1/y; + if (y.is_equal(_ex1)) + return 1/x; if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) { // treat all problematic x and y that may not be passed into tgamma, // because they would throw there although beta(x,y) is well-defined // using the formula beta(x,y) == (-1)^y * beta(1-x-y, y) - const numeric nx = ex_to(x); - const numeric ny = ex_to(y); + const numeric &nx = ex_to(x); + const numeric &ny = ex_to(y); if (nx.is_real() && nx.is_integer() && ny.is_real() && ny.is_integer()) { if (nx.is_negative()) { if (nx<=-ny) - return pow(_num_1(), ny)*beta(1-x-y, y); + return pow(*_num_1_p, ny)*beta(1-x-y, y); else throw (pole_error("beta_eval(): simple pole",1)); } if (ny.is_negative()) { if (ny<=-nx) - return pow(_num_1(), nx)*beta(1-y-x, x); + return pow(*_num_1_p, nx)*beta(1-y-x, x); else throw (pole_error("beta_eval(): simple pole",1)); } @@ -256,8 +286,9 @@ static ex beta_eval(const ex & x, const ex & y) if ((nx+ny).is_real() && (nx+ny).is_integer() && !(nx+ny).is_positive()) - return _ex0(); - // beta_evalf should be called here once it becomes available + return _ex0; + if (!ex_to(x).is_rational() || !ex_to(x).is_rational()) + return evalf(beta(x, y).hold()); } return beta(x,y).hold(); @@ -289,29 +320,29 @@ static ex beta_series(const ex & arg1, // Taylor series where there is no pole of one of the tgamma functions // falls back to beta function evaluation. Otherwise, fall back to // tgamma series directly. - const ex arg1_pt = arg1.subs(rel); - const ex arg2_pt = arg2.subs(rel); - GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol)); - const symbol *s = static_cast(rel.lhs().bp); + const ex arg1_pt = arg1.subs(rel, subs_options::no_pattern); + const ex arg2_pt = arg2.subs(rel, subs_options::no_pattern); + GINAC_ASSERT(is_a(rel.lhs())); + const symbol &s = ex_to(rel.lhs()); ex arg1_ser, arg2_ser, arg1arg2_ser; if ((!arg1_pt.info(info_flags::integer) || arg1_pt.info(info_flags::positive)) && (!arg2_pt.info(info_flags::integer) || arg2_pt.info(info_flags::positive))) throw do_taylor(); // caught by function::series() // trap the case where arg1 is on a pole: if (arg1.info(info_flags::integer) && !arg1.info(info_flags::positive)) - arg1_ser = tgamma(arg1+*s).series(rel, order, options); + arg1_ser = tgamma(arg1+s); else - arg1_ser = tgamma(arg1).series(rel,order); + arg1_ser = tgamma(arg1); // trap the case where arg2 is on a pole: if (arg2.info(info_flags::integer) && !arg2.info(info_flags::positive)) - arg2_ser = tgamma(arg2+*s).series(rel, order, options); + arg2_ser = tgamma(arg2+s); else - arg2_ser = tgamma(arg2).series(rel,order); + arg2_ser = tgamma(arg2); // trap the case where arg1+arg2 is on a pole: if ((arg1+arg2).info(info_flags::integer) && !(arg1+arg2).info(info_flags::positive)) - arg1arg2_ser = tgamma(arg2+arg1+*s).series(rel, order, options); + arg1arg2_ser = tgamma(arg2+arg1+s); else - arg1arg2_ser = tgamma(arg2+arg1).series(rel,order); + arg1arg2_ser = tgamma(arg2+arg1); // compose the result (expanding all the terms): return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand(); } @@ -321,8 +352,8 @@ REGISTER_FUNCTION(beta, eval_func(beta_eval). evalf_func(beta_evalf). derivative_func(beta_deriv). series_func(beta_series). - latex_name("\\mbox{B}"). - set_symmetry(sy_symm(0, 1))); + latex_name("\\mathrm{B}"). + set_symmetry(sy_symm(0, 1))); ////////// @@ -345,13 +376,13 @@ static ex psi1_evalf(const ex & x) static ex psi1_eval(const ex & x) { if (x.info(info_flags::numeric)) { - const numeric nx = ex_to(x); + const numeric &nx = ex_to(x); if (nx.is_integer()) { // integer case if (nx.is_positive()) { // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - Euler numeric rat = 0; - for (numeric i(nx+_num_1()); i>0; --i) + for (numeric i(nx+(*_num_1_p)); i>0; --i) rat += i.inverse(); return rat-Euler; } else { @@ -359,14 +390,14 @@ static ex psi1_eval(const ex & x) throw (pole_error("psi_eval(): simple pole",1)); } } - if ((_num2()*nx).is_integer()) { + if (((*_num2_p)*nx).is_integer()) { // half integer case if (nx.is_positive()) { // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - Euler - 2log(2) numeric rat = 0; - for (numeric i = (nx+_num_1())*_num2(); i>0; i-=_num2()) - rat += _num2()*i.inverse(); - return rat-Euler-_ex2()*log(_ex2()); + for (numeric i = (nx+(*_num_1_p))*(*_num2_p); i>0; i-=(*_num2_p)) + rat += (*_num2_p)*i.inverse(); + return rat-Euler-_ex2*log(_ex2); } else { // use the recurrence relation // psi(-m-1/2) == psi(-m-1/2+1) - 1 / (-m-1/2) @@ -375,8 +406,8 @@ static ex psi1_eval(const ex & x) // where r == ((-1/2)^(-1) + ... + (-m-1/2)^(-1)) numeric recur = 0; for (numeric p = nx; p<0; ++p) - recur -= pow(p, _num_1()); - return recur+psi(_ex1_2()); + recur -= pow(p, *_num_1_p); + return recur+psi(_ex1_2); } } // psi1_evalf should be called here once it becomes available @@ -390,7 +421,7 @@ static ex psi1_deriv(const ex & x, unsigned deriv_param) GINAC_ASSERT(deriv_param==0); // d/dx psi(x) -> psi(1,x) - return psi(_ex1(), x); + return psi(_ex1, x); } static ex psi1_series(const ex & arg, @@ -406,19 +437,19 @@ static ex psi1_series(const ex & arg, // from which follows // series(psi(x),x==-m,order) == // series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x==-m,order); - const ex arg_pt = arg.subs(rel); + const ex arg_pt = arg.subs(rel, subs_options::no_pattern); if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive)) throw do_taylor(); // caught by function::series() // if we got here we have to care for a simple pole at -m: const numeric m = -ex_to(arg_pt); ex recur; for (numeric p; p<=m; ++p) - recur += power(arg+p,_ex_1()); - return (psi(arg+m+_ex1())-recur).series(rel, order, options); + recur += power(arg+p,_ex_1); + return (psi(arg+m+_ex1)-recur).series(rel, order, options); } -const unsigned function_index_psi1 = - function::register_new(function_options("psi"). +unsigned psi1_SERIAL::serial = + function::register_new(function_options("psi", 1). eval_func(psi1_eval). evalf_func(psi1_evalf). derivative_func(psi1_deriv). @@ -449,17 +480,17 @@ static ex psi2_eval(const ex & n, const ex & x) if (n.is_zero()) return psi(x); // psi(-1,x) -> log(tgamma(x)) - if (n.is_equal(_ex_1())) + if (n.is_equal(_ex_1)) return log(tgamma(x)); if (n.info(info_flags::numeric) && n.info(info_flags::posint) && x.info(info_flags::numeric)) { - const numeric nn = ex_to(n); - const numeric nx = ex_to(x); + const numeric &nn = ex_to(n); + const numeric &nx = ex_to(x); if (nx.is_integer()) { // integer case - if (nx.is_equal(_num1())) + if (nx.is_equal(*_num1_p)) // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1) - return pow(_num_1(),nn+_num1())*factorial(nn)*zeta(ex(nn+_num1())); + return pow(*_num_1_p,nn+(*_num1_p))*factorial(nn)*zeta(ex(nn+(*_num1_p))); if (nx.is_positive()) { // use the recurrence relation // psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1) @@ -468,25 +499,25 @@ static ex psi2_eval(const ex & n, const ex & x) // where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1)) numeric recur = 0; for (numeric p = 1; p psi(n+1,x) - return psi(n+_ex1(), x); + return psi(n+_ex1, x); } static ex psi2_series(const ex & n, @@ -533,20 +564,20 @@ static ex psi2_series(const ex & n, // series(psi(x),x==-m,order) == // series(psi(x+m+1) - (-1)^n * n! * ((x)^(-n-1) + (x+1)^(-n-1) + ... // ... + (x+m)^(-n-1))),x==-m,order); - const ex arg_pt = arg.subs(rel); + const ex arg_pt = arg.subs(rel, subs_options::no_pattern); if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive)) throw do_taylor(); // caught by function::series() // if we got here we have to care for a pole of order n+1 at -m: const numeric m = -ex_to(arg_pt); ex recur; for (numeric p; p<=m; ++p) - recur += power(arg+p,-n+_ex_1()); - recur *= factorial(n)*power(_ex_1(),n); - return (psi(n, arg+m+_ex1())-recur).series(rel, order, options); + recur += power(arg+p,-n+_ex_1); + recur *= factorial(n)*power(_ex_1,n); + return (psi(n, arg+m+_ex1)-recur).series(rel, order, options); } -const unsigned function_index_psi2 = - function::register_new(function_options("psi"). +unsigned psi2_SERIAL::serial = + function::register_new(function_options("psi", 2). eval_func(psi2_eval). evalf_func(psi2_evalf). derivative_func(psi2_deriv).