* some related stuff. */
/*
- * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2003 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
#include "numeric.h"
#include "power.h"
#include "relational.h"
+#include "operators.h"
#include "symbol.h"
#include "symmetry.h"
#include "utils.h"
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));
}
// 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):
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);
}
{
if (x.info(info_flags::numeric)) {
// trap integer arguments:
- const numeric two_x = _num2()*ex_to<numeric>(x);
+ const numeric two_x = _num2*ex_to<numeric>(x);
if (two_x.is_even()) {
// tgamma(n) -> (n-1)! for postitive n
if (two_x.is_positive()) {
- return factorial(ex_to<numeric>(x).sub(_num1()));
+ return factorial(ex_to<numeric>(x).sub(_num1));
} else {
throw (pole_error("tgamma_eval(): simple pole",1));
}
// 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<numeric>(x).sub(_num1_2());
- return (doublefactorial(n.mul(_num2()).sub(_num1())).div(pow(_num2(),n))) * pow(Pi,_ex1_2());
+ const numeric n = ex_to<numeric>(x).sub(_num1_2);
+ return (doublefactorial(n.mul(_num2).sub(_num1)).div(pow(_num2,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<numeric>(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<numeric>(x).sub(_num1_2));
+ return (pow(_num_2, n).div(doublefactorial(n.mul(_num2).sub(_num1))))*sqrt(Pi);
}
}
// tgamma_evalf should be called here once it becomes available
// 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);
+ 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<numeric>(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+1, options);
}
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<numeric>(x);
- const numeric ny = ex_to<numeric>(y);
+ const numeric &nx = ex_to<numeric>(x);
+ const numeric &ny = ex_to<numeric>(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, 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, nx)*beta(1-y-x, x);
else
throw (pole_error("beta_eval(): simple pole",1));
}
if ((nx+ny).is_real() &&
(nx+ny).is_integer() &&
!(nx+ny).is_positive())
- return _ex0();
+ return _ex0;
// beta_evalf should be called here once it becomes available
}
// 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_exactly_a<symbol>(rel.lhs()));
+ 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<symbol>(rel.lhs()));
const symbol &s = ex_to<symbol>(rel.lhs());
ex arg1_ser, arg2_ser, arg1arg2_ser;
if ((!arg1_pt.info(info_flags::integer) || arg1_pt.info(info_flags::positive)) &&
static ex psi1_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
- const numeric nx = ex_to<numeric>(x);
+ const numeric &nx = ex_to<numeric>(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); i>0; --i)
rat += i.inverse();
return rat-Euler;
} else {
throw (pole_error("psi_eval(): simple pole",1));
}
}
- if ((_num2()*nx).is_integer()) {
+ if ((_num2*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)*_num2; i>0; i-=_num2)
+ rat += _num2*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)
// 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);
+ return recur+psi(_ex1_2);
}
}
// psi1_evalf should be called here once it becomes available
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,
// 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<numeric>(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 =
+unsigned psi1_SERIAL::serial =
function::register_new(function_options("psi").
eval_func(psi1_eval).
evalf_func(psi1_evalf).
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<numeric>(n);
- const numeric nx = ex_to<numeric>(x);
+ const numeric &nn = ex_to<numeric>(n);
+ const numeric &nx = ex_to<numeric>(x);
if (nx.is_integer()) {
// integer case
- if (nx.is_equal(_num1()))
+ if (nx.is_equal(_num1))
// 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,nn+_num1)*factorial(nn)*zeta(ex(nn+_num1));
if (nx.is_positive()) {
// use the recurrence relation
// psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1)
// where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1))
numeric recur = 0;
for (numeric p = 1; p<nx; ++p)
- recur += pow(p, -nn+_num_1());
- recur *= factorial(nn)*pow(_num_1(), nn);
- return recur+psi(n,_ex1());
+ recur += pow(p, -nn+_num_1);
+ recur *= factorial(nn)*pow(_num_1, nn);
+ return recur+psi(n,_ex1);
} else {
// for non-positive integers there is a pole:
throw (pole_error("psi2_eval(): pole",1));
}
}
- if ((_num2()*nx).is_integer()) {
+ if ((_num2*nx).is_integer()) {
// half integer case
- if (nx.is_equal(_num1_2()))
+ if (nx.is_equal(_num1_2))
// use psi(n,1/2) == (-)^(n+1) * n! * (2^(n+1)-1) * zeta(n+1)
- return pow(_num_1(),nn+_num1())*factorial(nn)*(pow(_num2(),nn+_num1()) + _num_1())*zeta(ex(nn+_num1()));
+ return pow(_num_1,nn+_num1)*factorial(nn)*(pow(_num2,nn+_num1) + _num_1)*zeta(ex(nn+_num1));
if (nx.is_positive()) {
- const numeric m = nx - _num1_2();
+ const numeric m = nx - _num1_2;
// use the multiplication formula
// psi(n,2*m) == (psi(n,m) + psi(n,m+1/2)) / 2^(n+1)
// to revert to positive integer case
- return psi(n,_num2()*m)*pow(_num2(),nn+_num1())-psi(n,m);
+ return psi(n,_num2*m)*pow(_num2,nn+_num1)-psi(n,m);
} else {
// use the recurrence relation
// psi(n,-m-1/2) == psi(n,-m-1/2+1) - (-)^n * n! / (-m-1/2)^(n+1)
// where r == (-)^(n+1) * n! * ((-1/2)^(-n-1) + ... + (-m-1/2)^(-n-1))
numeric recur = 0;
for (numeric p = nx; p<0; ++p)
- recur += pow(p, -nn+_num_1());
- recur *= factorial(nn)*pow(_num_1(), nn+_num_1());
- return recur+psi(n,_ex1_2());
+ recur += pow(p, -nn+_num_1);
+ recur *= factorial(nn)*pow(_num_1, nn+_num_1);
+ return recur+psi(n,_ex1_2);
}
}
// psi2_evalf should be called here once it becomes available
throw(std::logic_error("cannot diff psi(n,x) with respect to n"));
}
// d/dx psi(n,x) -> psi(n+1,x)
- return psi(n+_ex1(), x);
+ return psi(n+_ex1, x);
}
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<numeric>(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 =
+unsigned psi2_SERIAL::serial =
function::register_new(function_options("psi").
eval_func(psi2_eval).
evalf_func(psi2_evalf).
git://www.ginac.de / ginac.git / blobdiff