Added series expansion for functions (classical) Li and S around x==0.
authorJens Vollinga <vollinga@thep.physik.uni-mainz.de>
Thu, 20 Oct 2005 13:15:32 +0000 (13:15 +0000)
committerJens Vollinga <vollinga@thep.physik.uni-mainz.de>
Thu, 20 Oct 2005 13:15:32 +0000 (13:15 +0000)
ginac/inifcns_nstdsums.cpp

index f5d33ec79cb15c54820d22f6bbac9fe9e00092d5..1fd80aefe2e453b269fcf34e403d9149669e2014 100644 (file)
@@ -1504,9 +1504,37 @@ static ex Li_eval(const ex& m_, const ex& x_)
 
 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
 {
-       epvector seq;
-       seq.push_back(expair(Li(m, x), 0));
-       return pseries(rel, seq);
+       if (is_a<lst>(m) || is_a<lst>(x)) {
+               // multiple polylog
+               epvector seq;
+               seq.push_back(expair(Li(m, x), 0));
+               return pseries(rel, seq);
+       }
+       
+       // classical polylog
+       const ex x_pt = x.subs(rel, subs_options::no_pattern);
+       if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
+               // First special case: x==0 (derivatives have poles)
+               if (x_pt.is_zero()) {
+                       const symbol s;
+                       ex ser;
+                       // manually construct the primitive expansion
+                       for (int i=1; i<order; ++i)
+                               ser += pow(s,i) / pow(numeric(i), m);
+                       // substitute the argument's series expansion
+                       ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
+                       // maybe that was terminating, so add a proper order term
+                       epvector nseq;
+                       nseq.push_back(expair(Order(_ex1), order));
+                       ser += pseries(rel, nseq);
+                       // reexpanding it will collapse the series again
+                       return ser.series(rel, order);
+               }
+               // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
+               throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
+       }
+       // all other cases should be safe, by now:
+       throw do_taylor();  // caught by function::series()
 }
 
 
@@ -1988,9 +2016,48 @@ static ex S_eval(const ex& n, const ex& p, const ex& x)
 
 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
 {
-       epvector seq;
-       seq.push_back(expair(S(n, p, x), 0));
-       return pseries(rel, seq);
+       if (p == _ex1) {
+               return Li(n+1, x).series(rel, order, options);
+       }
+
+       const ex x_pt = x.subs(rel, subs_options::no_pattern);
+       if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
+               // First special case: x==0 (derivatives have poles)
+               if (x_pt.is_zero()) {
+                       const symbol s;
+                       ex ser;
+                       // manually construct the primitive expansion
+                       // subsum = Euler-Zagier-Sum is needed
+                       // dirty hack (slow ...) calculation of subsum:
+                       std::vector<ex> presubsum, subsum;
+                       subsum.push_back(0);
+                       for (int i=1; i<order-1; ++i) {
+                               subsum.push_back(subsum[i-1] + numeric(1, i));
+                       }
+                       for (int depth=2; depth<p; ++depth) {
+                               presubsum = subsum;
+                               for (int i=1; i<order-1; ++i) {
+                                       subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
+                               }
+                       }
+                               
+                       for (int i=1; i<order; ++i) {
+                               ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
+                       }
+                       // substitute the argument's series expansion
+                       ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
+                       // maybe that was terminating, so add a proper order term
+                       epvector nseq;
+                       nseq.push_back(expair(Order(_ex1), order));
+                       ser += pseries(rel, nseq);
+                       // reexpanding it will collapse the series again
+                       return ser.series(rel, order);
+               }
+               // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
+               throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
+       }
+       // all other cases should be safe, by now:
+       throw do_taylor();  // caught by function::series()
 }