static ex Li2_eval(const ex & x)
{
+ // Li2(0) -> 0
if (x.is_zero())
return x;
+ // Li2(1) -> Pi^2/6
if (x.is_equal(_ex1()))
- return power(Pi, _ex2()) / _ex6();
+ return power(Pi,_ex2())/_ex6();
+ // Li2(1/2) -> Pi^2/12 - log(2)^2/2
+ if (x.is_equal(_ex1_2()))
+ return power(Pi,_ex2())/_ex12() + power(log(_ex2()),_ex2())*_ex_1_2();
+ // Li2(-1) -> -Pi^2/12
if (x.is_equal(_ex_1()))
- return -power(Pi, _ex2()) / _ex12();
+ return -power(Pi,_ex2())/_ex12();
+ // Li2(I) -> -Pi^2/48+Catalan*I
+ if (x.is_equal(I))
+ return power(Pi,_ex2())/_ex_48() + Catalan*I;
+ // Li2(-I) -> -Pi^2/48-Catalan*I
+ if (x.is_equal(-I))
+ return power(Pi,_ex2())/_ex_48() - Catalan*I;
return Li2(x).hold();
}
-REGISTER_FUNCTION(Li2, eval_func(Li2_eval));
+static ex Li2_deriv(const ex & x, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param==0);
+
+ // d/dx Li2(x) -> -log(1-x)/x
+ return -log(1-x)/x;
+}
+
+static ex Li2_series(const ex &x, const relational &rel, int order)
+{
+ const ex x_pt = x.subs(rel);
+ if (!x_pt.is_zero() && !x_pt.is_equal(_ex1()))
+ throw do_taylor(); // caught by function::series()
+ // First case: x==0 (derivatives have poles)
+ if (x_pt.is_zero()) {
+ // method:
+ // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
+ // simply substitute x==0. The limit, however, exists: it is 1. We
+ // also know all higher derivatives' limits: (d/dx)^n Li2(x) == n!/n^2.
+ // So the primitive series expansion is Li2(x==0) == x + x^2/4 + x^3/9
+ // and so on.
+ // We first construct such a primitive series expansion manually in
+ // a dummy symbol s and then insert the argument's series expansion
+ // for s. Reexpanding the resulting series returns the desired result.
+ const symbol s;
+ ex ser;
+ // construct manually the primitive expansion
+ for (int i=1; i<order; ++i)
+ ser += pow(s,i)/pow(numeric(i),numeric(2));
+ // substitute the argument's series expansion
+ ser = ser.subs(s==x.series(rel,order));
+ // maybe that was terminanting, so add a proper order term
+ epvector nseq;
+ nseq.push_back(expair(Order(_ex1()), numeric(order)));
+ ser += pseries(rel, nseq);
+ // reexpand will collapse the series again
+ ser = ser.series(rel,order);
+ return ser;
+ }
+ // second problematic case: x real, >=1 (branch cut)
+ return pseries();
+ // TODO: Li2_series should do something around branch point?
+ // Careful: may involve logs!
+}
+
+REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
+ derivative_func(Li2_deriv).
+ series_func(Li2_series));
//////////
// trilogarithm
// `construct on first use' chest of numbers
//////////
+// numeric -120
+const numeric & _num_120(void)
+{
+ const static ex e = ex(numeric(-120));
+ const static numeric * n = static_cast<const numeric *>(e.bp);
+ return *n;
+}
+
+const ex & _ex_120(void)
+{
+ static ex * e = new ex(_num_120());
+ return *e;
+}
+
// numeric -60
const numeric & _num_60(void)
{
return *e;
}
-// numeric -120
-const numeric & _num_120(void)
+// numeric -48
+const numeric & _num_48(void)
{
- const static ex e = ex(numeric(-120));
+ const static ex e = ex(numeric(-48));
const static numeric * n = static_cast<const numeric *>(e.bp);
return *n;
}
-const ex & _ex_120(void)
+const ex & _ex_48(void)
{
- static ex * e = new ex(_num_120());
+ static ex * e = new ex(_num_48());
return *e;
}
return *e;
}
+// numeric 48
+const numeric & _num48(void)
+{
+ const static ex e = ex(numeric(48));
+ const static numeric * n = static_cast<const numeric *>(e.bp);
+ return *n;
+}
+
+const ex & _ex48(void)
+{
+ static ex * e = new ex(_num48());
+ return *e;
+}
+
// numeric 60
const numeric & _num60(void)
{
const ex & _ex_120(void);
const numeric & _num_60(void); // -60
const ex & _ex_60(void);
+const numeric & _num_48(void); // -48
+const ex & _ex_48(void);
const numeric & _num_30(void); // -30
const ex & _ex_30(void);
const numeric & _num_25(void); // -25
const ex & _ex25(void);
const numeric & _num30(void); // 30
const ex & _ex30(void);
+const numeric & _num48(void); // 48
+const ex & _ex48(void);
const numeric & _num60(void); // 60
const ex & _ex60(void);
const numeric & _num120(void); // 120