return coefficient*power(Pi,_ex1_2());
}
}
+ // gamma_evalf should be called here once it becomes available
}
+
return gamma(x).hold();
}
(!ypoint.info(info_flags::integer) || ypoint.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:
- throw (std::domain_error("beta_series(): please code me"));
- /*numeric m = -ex_to_numeric(xpoint);
- *ex ser_numer = gamma(x+m+_ex1());
- *ex ser_denom = _ex1();
- *for (numeric p; p<=m; ++p)
- * ser_denom *= x+p;
- *return (ser_numer/ser_denom).series(s, point, order+1);*/
+ throw (std::domain_error("beta_series(): Mama, please code me!"));
}
REGISTER_FUNCTION(beta, beta_eval, beta_evalf, beta_diff, beta_series);
static ex psi1_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
- if (x.info(info_flags::integer) && !x.info(info_flags::positive))
- throw (std::domain_error("psi_eval(): simple pole"));
- if (x.info(info_flags::positive)) {
- // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - EulerGamma
- if (x.info(info_flags::integer)) {
+ numeric nx = ex_to_numeric(x);
+ if (nx.is_integer()) {
+ // integer case
+ if (nx.is_positive()) {
+ // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - EulerGamma
numeric rat(0);
- for (numeric i(ex_to_numeric(x)-_num1()); i.is_positive(); --i)
+ for (numeric i(nx+_num_1()); i.is_positive(); --i)
rat += i.inverse();
return rat-EulerGamma;
+ } else {
+ // for non-positive integers there is a pole:
+ throw (std::domain_error("psi_eval(): simple pole"));
}
- // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - EulerGamma - 2log(2)
- if ((_ex2()*x).info(info_flags::integer)) {
+ }
+ if ((_num2()*nx).is_integer()) {
+ // half integer case
+ if (nx.is_positive()) {
+ // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - EulerGamma - 2log(2)
numeric rat(0);
- for (numeric i((ex_to_numeric(x)-_num1())*_num2()); i.is_positive(); i-=_num2())
- rat += _num2()*i.inverse();
- return rat-EulerGamma-_ex2()*log(_ex2());
- }
- if (x.compare(_ex1())==1) {
- // should call numeric, since >1
+ for (numeric i((nx+_num_1())*_num2()); i.is_positive(); i-=_num2())
+ rat += _num2()*i.inverse();
+ return rat-EulerGamma-_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))
+ numeric recur(0);
+ for (numeric p(nx); p<0; ++p)
+ recur -= pow(p, _num_1());
+ return recur+psi(_ex1_2());
}
}
+ // psi1_evalf should be called here once it becomes available
}
+
return psi(x).hold();
}
numeric nn = ex_to_numeric(n);
numeric nx = ex_to_numeric(x);
if (nx.is_integer()) {
+ // integer case
if (nx.is_equal(_num1()))
- return pow(_num_1(), nn+_num1())*factorial(nn)*zeta(ex(nn+_num1()));
+ // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1)
+ 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)
// 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;
+ 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 (std::domain_error("psi2_eval(): pole"));
}
- // for non-positive integers there is a pole:
- throw (std::domain_error("psi2_eval(): pole"));
}
+ if ((_num2()*nx).is_integer()) {
+ // half integer case
+ 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()));
+ if (nx.is_positive()) {
+ 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);
+ } 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());
+ }
+ }
+ // psi2_evalf should be called here once it becomes available
}
+
return psi(n, x).hold();
}