*
* 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
*
* 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
if (x.info(info_flags::integer)) {
// lgamma(n) -> log((n-1)!) for postitive n
if (x.info(info_flags::posint))
if (x.info(info_flags::integer)) {
// lgamma(n) -> log((n-1)!) for postitive n
if (x.info(info_flags::posint))
// from which follows
// series(lgamma(x),x==-m,order) ==
// series(lgamma(x+m+1)-log(x)...-log(x+m)),x==-m,order);
// from which follows
// series(lgamma(x),x==-m,order) ==
// series(lgamma(x+m+1)-log(x)...-log(x+m)),x==-m,order);
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):
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):
} else {
throw (pole_error("tgamma_eval(): simple pole",1));
}
}
// trap half integer arguments:
} else {
throw (pole_error("tgamma_eval(): simple pole",1));
}
}
// trap half integer arguments:
- if ((x*_ex2()).info(info_flags::posint)) {
- numeric n = ex_to_numeric(x).sub(_num1_2());
- numeric coefficient = doublefactorial(n.mul(_num2()).sub(_num1()));
- coefficient = coefficient.div(pow(_num2(),n));
- return coefficient * pow(Pi,_ex1_2());
+ 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))) * sqrt(Pi);
- numeric n = abs(ex_to_numeric(x).sub(_num1_2()));
- numeric coefficient = pow(_num_2(), n);
- coefficient = coefficient.div(doublefactorial(n.mul(_num2()).sub(_num1())));;
- return coefficient*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(x) == tgamma(x+1) / x
// from which follows
// series(tgamma(x),x==-m,order) ==
// 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:
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:
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- TYPECHECK(y,numeric)
- END_TYPECHECK(beta(x,y))
+ if (is_exactly_a<numeric>(x) && is_exactly_a<numeric>(y)) {
+ try {
+ return tgamma(ex_to<numeric>(x))*tgamma(ex_to<numeric>(y))/tgamma(ex_to<numeric>(x+y));
+ } catch (const dunno &e) { }
+ }
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)
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)
- (nx+ny).is_integer() &&
- !(nx+ny).is_positive())
- return _ex0();
- // everything is ok:
- return tgamma(x)*tgamma(y)/tgamma(x+y);
+ (nx+ny).is_integer() &&
+ !(nx+ny).is_positive())
+ 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.
// 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<symbol *>(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<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)) &&
(!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))
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))
// trap the case where arg2 is on a pole:
if (arg2.info(info_flags::integer) && !arg2.info(info_flags::positive))
// trap the case where arg2 is on a pole:
if (arg2.info(info_flags::integer) && !arg2.info(info_flags::positive))
// trap the case where arg1+arg2 is on a pole:
if ((arg1+arg2).info(info_flags::integer) && !(arg1+arg2).info(info_flags::positive))
// trap the case where arg1+arg2 is on a pole:
if ((arg1+arg2).info(info_flags::integer) && !(arg1+arg2).info(info_flags::positive))
// compose the result (expanding all the terms):
return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand();
}
// compose the result (expanding all the terms):
return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand();
}
- numeric rat(0);
- for (numeric i((nx+_num_1())*_num2()); i.is_positive(); i-=_num2())
- rat += _num2()*i.inverse();
- return rat-Euler-_ex2()*log(_ex2());
+ numeric rat = 0;
+ 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)
// to relate psi(-m-1/2) to psi(1/2):
// psi(-m-1/2) == psi(1/2) + r
// where r == ((-1/2)^(-1) + ... + (-m-1/2)^(-1))
} else {
// use the recurrence relation
// psi(-m-1/2) == psi(-m-1/2+1) - 1 / (-m-1/2)
// to relate psi(-m-1/2) to psi(1/2):
// psi(-m-1/2) == psi(1/2) + r
// 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());
+ numeric recur = 0;
+ for (numeric p = nx; p<0; ++p)
+ recur -= pow(p, _num_1);
+ return recur+psi(_ex1_2);
GINAC_ASSERT(deriv_param==0);
// d/dx psi(x) -> psi(1,x)
GINAC_ASSERT(deriv_param==0);
// d/dx psi(x) -> psi(1,x)
// 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);
// 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);
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:
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:
- 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).
- BEGIN_TYPECHECK
- TYPECHECK(n,numeric)
- TYPECHECK(x,numeric)
- END_TYPECHECK(psi(n,x))
+ if (is_exactly_a<numeric>(n) && is_exactly_a<numeric>(x)) {
+ try {
+ return psi(ex_to<numeric>(n),ex_to<numeric>(x));
+ } catch (const dunno &e) { }
+ }
if (nx.is_positive()) {
// use the recurrence relation
// psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1)
// to relate psi(n,m) to psi(n,1):
// psi(n,m) == psi(n,1) + r
// where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1))
if (nx.is_positive()) {
// use the recurrence relation
// psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1)
// to relate psi(n,m) to psi(n,1):
// psi(n,m) == psi(n,1) + r
// 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());
+ 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);
} else {
// for non-positive integers there is a pole:
throw (pole_error("psi2_eval(): pole",1));
}
}
} else {
// for non-positive integers there is a pole:
throw (pole_error("psi2_eval(): pole",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));
// 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
// 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)
// to relate psi(n,-m-1/2) to psi(n,1/2):
// psi(n,-m-1/2) == psi(n,1/2) + r
// where r == (-)^(n+1) * n! * ((-1/2)^(-n-1) + ... + (-m-1/2)^(-n-1))
} else {
// use the recurrence relation
// psi(n,-m-1/2) == psi(n,-m-1/2+1) - (-)^n * n! / (-m-1/2)^(n+1)
// to relate psi(n,-m-1/2) to psi(n,1/2):
// psi(n,-m-1/2) == psi(n,1/2) + r
// 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());
+ 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);
throw(std::logic_error("cannot diff psi(n,x) with respect to n"));
}
// d/dx psi(n,x) -> psi(n+1,x)
throw(std::logic_error("cannot diff psi(n,x) with respect to n"));
}
// d/dx psi(n,x) -> psi(n+1,x)
// 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);
// 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);
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:
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:
- 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).