1 /** @file inifcns_nstdsums.cpp
3 * Implementation of some special functions that have a representation as nested sums.
6 * classical polylogarithm Li(n,x)
7 * multiple polylogarithm Li(lst(m_1,...,m_k),lst(x_1,...,x_k))
8 * G(lst(a_1,...,a_k),y) or G(lst(a_1,...,a_k),lst(s_1,...,s_k),y)
9 * Nielsen's generalized polylogarithm S(n,p,x)
10 * harmonic polylogarithm H(m,x) or H(lst(m_1,...,m_k),x)
11 * multiple zeta value zeta(m) or zeta(lst(m_1,...,m_k))
12 * alternating Euler sum zeta(m,s) or zeta(lst(m_1,...,m_k),lst(s_1,...,s_k))
16 * - All formulae used can be looked up in the following publications:
17 * [Kol] Nielsen's Generalized Polylogarithms, K.S.Kolbig, SIAM J.Math.Anal. 17 (1986), pp. 1232-1258.
18 * [Cra] Fast Evaluation of Multiple Zeta Sums, R.E.Crandall, Math.Comp. 67 (1998), pp. 1163-1172.
19 * [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
20 * [BBB] Special Values of Multiple Polylogarithms, J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
21 * [VSW] Numerical evaluation of multiple polylogarithms, J.Vollinga, S.Weinzierl, hep-ph/0410259
23 * - The order of parameters and arguments of Li and zeta is defined according to the nested sums
24 * representation. The parameters for H are understood as in [ReV]. They can be in expanded --- only
25 * 0, 1 and -1 --- or in compactified --- a string with zeros in front of 1 or -1 is written as a single
26 * number --- notation.
28 * - All functions can be nummerically evaluated with arguments in the whole complex plane. The parameters
29 * for Li, zeta and S must be positive integers. If you want to have an alternating Euler sum, you have
30 * to give the signs of the parameters as a second argument s to zeta(m,s) containing 1 and -1.
32 * - The calculation of classical polylogarithms is speeded up by using Bernoulli numbers and
33 * look-up tables. S uses look-up tables as well. The zeta function applies the algorithms in
34 * [Cra] and [BBB] for speed up. Multiple polylogarithms use Hoelder convolution [BBB].
36 * - The functions have no means to do a series expansion into nested sums. To do this, you have to convert
37 * these functions into the appropriate objects from the nestedsums library, do the expansion and convert
40 * - Numerical testing of this implementation has been performed by doing a comparison of results
41 * between this software and the commercial M.......... 4.1. Multiple zeta values have been checked
42 * by means of evaluations into simple zeta values. Harmonic polylogarithms have been checked by
43 * comparison to S(n,p,x) for corresponding parameter combinations and by continuity checks
44 * around |x|=1 along with comparisons to corresponding zeta functions. Multiple polylogarithms were
45 * checked against H and zeta and by means of shuffle and quasi-shuffle relations.
50 * GiNaC Copyright (C) 1999-2007 Johannes Gutenberg University Mainz, Germany
52 * This program is free software; you can redistribute it and/or modify
53 * it under the terms of the GNU General Public License as published by
54 * the Free Software Foundation; either version 2 of the License, or
55 * (at your option) any later version.
57 * This program is distributed in the hope that it will be useful,
58 * but WITHOUT ANY WARRANTY; without even the implied warranty of
59 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
60 * GNU General Public License for more details.
62 * You should have received a copy of the GNU General Public License
63 * along with this program; if not, write to the Free Software
64 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
79 #include "operators.h"
82 #include "relational.h"
91 //////////////////////////////////////////////////////////////////////
93 // Classical polylogarithm Li(n,x)
97 //////////////////////////////////////////////////////////////////////
100 // anonymous namespace for helper functions
104 // lookup table for factors built from Bernoulli numbers
106 std::vector<std::vector<cln::cl_N> > Xn;
107 // initial size of Xn that should suffice for 32bit machines (must be even)
108 const int xninitsizestep = 26;
109 int xninitsize = xninitsizestep;
113 // This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms.
114 // With these numbers the polylogs can be calculated as follows:
115 // Li_p (x) = \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with u = -log(1-x)
116 // X_0(n) = B_n (Bernoulli numbers)
117 // X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k)
118 // The calculation of Xn depends on X0 and X{n-1}.
119 // X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater.
120 // This results in a slightly more complicated algorithm for the X_n.
121 // The first index in Xn corresponds to the index of the polylog minus 2.
122 // The second index in Xn corresponds to the index from the actual sum.
126 // calculate X_2 and higher (corresponding to Li_4 and higher)
127 std::vector<cln::cl_N> buf(xninitsize);
128 std::vector<cln::cl_N>::iterator it = buf.begin();
130 *it = -(cln::expt(cln::cl_I(2),n+1) - 1) / cln::expt(cln::cl_I(2),n+1); // i == 1
132 for (int i=2; i<=xninitsize; i++) {
134 result = 0; // k == 0
136 result = Xn[0][i/2-1]; // k == 0
138 for (int k=1; k<i-1; k++) {
139 if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
140 result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
143 result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
144 result = result + Xn[n-1][i-1] / (i+1); // k == i
151 // special case to handle the X_0 correct
152 std::vector<cln::cl_N> buf(xninitsize);
153 std::vector<cln::cl_N>::iterator it = buf.begin();
155 *it = cln::cl_I(-3)/cln::cl_I(4); // i == 1
157 *it = cln::cl_I(17)/cln::cl_I(36); // i == 2
159 for (int i=3; i<=xninitsize; i++) {
161 result = -Xn[0][(i-3)/2]/2;
162 *it = (cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result;
165 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
166 for (int k=1; k<i/2; k++) {
167 result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
176 std::vector<cln::cl_N> buf(xninitsize/2);
177 std::vector<cln::cl_N>::iterator it = buf.begin();
178 for (int i=1; i<=xninitsize/2; i++) {
179 *it = bernoulli(i*2).to_cl_N();
188 // doubles the number of entries in each Xn[]
191 const int pos0 = xninitsize / 2;
193 for (int i=1; i<=xninitsizestep/2; ++i) {
194 Xn[0].push_back(bernoulli((i+pos0)*2).to_cl_N());
197 int xend = xninitsize + xninitsizestep;
200 for (int i=xninitsize+1; i<=xend; ++i) {
202 result = -Xn[0][(i-3)/2]/2;
203 Xn[1].push_back((cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result);
205 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
206 for (int k=1; k<i/2; k++) {
207 result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
209 Xn[1].push_back(result);
213 for (int n=2; n<Xn.size(); ++n) {
214 for (int i=xninitsize+1; i<=xend; ++i) {
216 result = 0; // k == 0
218 result = Xn[0][i/2-1]; // k == 0
220 for (int k=1; k<i-1; ++k) {
221 if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
222 result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
225 result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
226 result = result + Xn[n-1][i-1] / (i+1); // k == i
227 Xn[n].push_back(result);
231 xninitsize += xninitsizestep;
235 // calculates Li(2,x) without Xn
236 cln::cl_N Li2_do_sum(const cln::cl_N& x)
240 cln::cl_N num = x * cln::cl_float(1, cln::float_format(Digits));
241 cln::cl_I den = 1; // n^2 = 1
246 den = den + i; // n^2 = 4, 9, 16, ...
248 res = res + num / den;
249 } while (res != resbuf);
254 // calculates Li(2,x) with Xn
255 cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x)
257 std::vector<cln::cl_N>::const_iterator it = Xn[0].begin();
258 std::vector<cln::cl_N>::const_iterator xend = Xn[0].end();
259 cln::cl_N u = -cln::log(1-x);
260 cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
261 cln::cl_N uu = cln::square(u);
262 cln::cl_N res = u - uu/4;
267 factor = factor * uu / (2*i * (2*i+1));
268 res = res + (*it) * factor;
272 it = Xn[0].begin() + (i-1);
275 } while (res != resbuf);
280 // calculates Li(n,x), n>2 without Xn
281 cln::cl_N Lin_do_sum(int n, const cln::cl_N& x)
283 cln::cl_N factor = x * cln::cl_float(1, cln::float_format(Digits));
290 res = res + factor / cln::expt(cln::cl_I(i),n);
292 } while (res != resbuf);
297 // calculates Li(n,x), n>2 with Xn
298 cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x)
300 std::vector<cln::cl_N>::const_iterator it = Xn[n-2].begin();
301 std::vector<cln::cl_N>::const_iterator xend = Xn[n-2].end();
302 cln::cl_N u = -cln::log(1-x);
303 cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
309 factor = factor * u / i;
310 res = res + (*it) * factor;
314 it = Xn[n-2].begin() + (i-2);
315 xend = Xn[n-2].end();
317 } while (res != resbuf);
322 // forward declaration needed by function Li_projection and C below
323 numeric S_num(int n, int p, const numeric& x);
326 // helper function for classical polylog Li
327 cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
329 // treat n=2 as special case
331 // check if precalculated X0 exists
336 if (cln::realpart(x) < 0.5) {
337 // choose the faster algorithm
338 // the switching point was empirically determined. the optimal point
339 // depends on hardware, Digits, ... so an approx value is okay.
340 // it solves also the problem with precision due to the u=-log(1-x) transformation
341 if (cln::abs(cln::realpart(x)) < 0.25) {
343 return Li2_do_sum(x);
345 return Li2_do_sum_Xn(x);
348 // choose the faster algorithm
349 if (cln::abs(cln::realpart(x)) > 0.75) {
350 return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
352 return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
356 // check if precalculated Xn exist
358 for (int i=xnsize; i<n-1; i++) {
363 if (cln::realpart(x) < 0.5) {
364 // choose the faster algorithm
365 // with n>=12 the "normal" summation always wins against the method with Xn
366 if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) {
367 return Lin_do_sum(n, x);
369 return Lin_do_sum_Xn(n, x);
372 cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
373 for (int j=0; j<n-1; j++) {
374 result = result + (S_num(n-j-1, 1, 1).to_cl_N() - S_num(1, n-j-1, 1-x).to_cl_N())
375 * cln::expt(cln::log(x), j) / cln::factorial(j);
383 // helper function for classical polylog Li
384 numeric Lin_numeric(int n, const numeric& x)
388 return -cln::log(1-x.to_cl_N());
399 return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
401 if (abs(x.real()) < 0.4 && abs(abs(x)-1) < 0.01) {
402 cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
403 cln::cl_N result = -cln::expt(cln::log(x_), n-1) * cln::log(1-x_) / cln::factorial(n-1);
404 for (int j=0; j<n-1; j++) {
405 result = result + (S_num(n-j-1, 1, 1).to_cl_N() - S_num(1, n-j-1, 1-x_).to_cl_N())
406 * cln::expt(cln::log(x_), j) / cln::factorial(j);
411 // what is the desired float format?
412 // first guess: default format
413 cln::float_format_t prec = cln::default_float_format;
414 const cln::cl_N value = x.to_cl_N();
415 // second guess: the argument's format
416 if (!x.real().is_rational())
417 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
418 else if (!x.imag().is_rational())
419 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
422 if (cln::abs(value) > 1) {
423 cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
424 // check if argument is complex. if it is real, the new polylog has to be conjugated.
425 if (cln::zerop(cln::imagpart(value))) {
427 result = result + conjugate(Li_projection(n, cln::recip(value), prec));
430 result = result - conjugate(Li_projection(n, cln::recip(value), prec));
435 result = result + Li_projection(n, cln::recip(value), prec);
438 result = result - Li_projection(n, cln::recip(value), prec);
442 for (int j=0; j<n-1; j++) {
443 add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
444 * Lin_numeric(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
446 result = result - add;
450 return Li_projection(n, value, prec);
455 } // end of anonymous namespace
458 //////////////////////////////////////////////////////////////////////
460 // Multiple polylogarithm Li(n,x)
464 //////////////////////////////////////////////////////////////////////
467 // anonymous namespace for helper function
471 // performs the actual series summation for multiple polylogarithms
472 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
474 // ensure all x <> 0.
475 for (std::vector<cln::cl_N>::const_iterator it = x.begin(); it != x.end(); ++it) {
476 if ( *it == 0 ) return cln::cl_float(0, cln::float_format(Digits));
479 const int j = s.size();
480 bool flag_accidental_zero = false;
482 std::vector<cln::cl_N> t(j);
483 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
490 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
491 for (int k=j-2; k>=0; k--) {
492 flag_accidental_zero = cln::zerop(t[k+1]);
493 t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]);
495 } while ( (t[0] != t0buf) || flag_accidental_zero );
501 // converts parameter types and calls multipleLi_do_sum (convenience function for G_numeric)
502 cln::cl_N mLi_do_summation(const lst& m, const lst& x)
504 std::vector<int> m_int;
505 std::vector<cln::cl_N> x_cln;
506 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
507 m_int.push_back(ex_to<numeric>(*itm).to_int());
508 x_cln.push_back(ex_to<numeric>(*itx).to_cl_N());
510 return multipleLi_do_sum(m_int, x_cln);
514 // forward declaration for Li_eval()
515 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
518 // holding dummy-symbols for the G/Li transformations
519 std::vector<ex> gsyms;
522 // type used by the transformation functions for G
523 typedef std::vector<int> Gparameter;
526 // G_eval1-function for G transformations
527 ex G_eval1(int a, int scale)
530 const ex& scs = gsyms[std::abs(scale)];
531 const ex& as = gsyms[std::abs(a)];
533 return -log(1 - scs/as);
538 return log(gsyms[std::abs(scale)]);
543 // G_eval-function for G transformations
544 ex G_eval(const Gparameter& a, int scale)
546 // check for properties of G
547 ex sc = gsyms[std::abs(scale)];
549 bool all_zero = true;
550 bool all_ones = true;
552 for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
554 const ex sym = gsyms[std::abs(*it)];
568 // care about divergent G: shuffle to separate divergencies that will be canceled
569 // later on in the transformation
570 if (newa.nops() > 1 && newa.op(0) == sc && !all_ones && a.front()!=0) {
573 Gparameter::const_iterator it = a.begin();
575 for (; it != a.end(); ++it) {
576 short_a.push_back(*it);
578 ex result = G_eval1(a.front(), scale) * G_eval(short_a, scale);
579 it = short_a.begin();
580 for (int i=1; i<count_ones; ++i) {
583 for (; it != short_a.end(); ++it) {
586 Gparameter::const_iterator it2 = short_a.begin();
587 for (--it2; it2 != it;) {
589 newa.push_back(*it2);
591 newa.push_back(a[0]);
593 for (; it2 != short_a.end(); ++it2) {
594 newa.push_back(*it2);
596 result -= G_eval(newa, scale);
598 return result / count_ones;
601 // G({1,...,1};y) -> G({1};y)^k / k!
602 if (all_ones && a.size() > 1) {
603 return pow(G_eval1(a.front(),scale), count_ones) / factorial(count_ones);
606 // G({0,...,0};y) -> log(y)^k / k!
608 return pow(log(gsyms[std::abs(scale)]), a.size()) / factorial(a.size());
611 // no special cases anymore -> convert it into Li
614 ex argbuf = gsyms[std::abs(scale)];
616 for (Gparameter::const_iterator it=a.begin(); it!=a.end(); ++it) {
618 const ex& sym = gsyms[std::abs(*it)];
619 x.append(argbuf / sym);
627 return pow(-1, x.nops()) * Li(m, x);
631 // converts data for G: pending_integrals -> a
632 Gparameter convert_pending_integrals_G(const Gparameter& pending_integrals)
634 GINAC_ASSERT(pending_integrals.size() != 1);
636 if (pending_integrals.size() > 0) {
637 // get rid of the first element, which would stand for the new upper limit
638 Gparameter new_a(pending_integrals.begin()+1, pending_integrals.end());
641 // just return empty parameter list
648 // check the parameters a and scale for G and return information about convergence, depth, etc.
649 // convergent : true if G(a,scale) is convergent
650 // depth : depth of G(a,scale)
651 // trailing_zeros : number of trailing zeros of a
652 // min_it : iterator of a pointing on the smallest element in a
653 Gparameter::const_iterator check_parameter_G(const Gparameter& a, int scale,
654 bool& convergent, int& depth, int& trailing_zeros, Gparameter::const_iterator& min_it)
660 Gparameter::const_iterator lastnonzero = a.end();
661 for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
662 if (std::abs(*it) > 0) {
666 if (std::abs(*it) < scale) {
668 if ((min_it == a.end()) || (std::abs(*it) < std::abs(*min_it))) {
676 return ++lastnonzero;
680 // add scale to pending_integrals if pending_integrals is empty
681 Gparameter prepare_pending_integrals(const Gparameter& pending_integrals, int scale)
683 GINAC_ASSERT(pending_integrals.size() != 1);
685 if (pending_integrals.size() > 0) {
686 return pending_integrals;
688 Gparameter new_pending_integrals;
689 new_pending_integrals.push_back(scale);
690 return new_pending_integrals;
695 // handles trailing zeroes for an otherwise convergent integral
696 ex trailing_zeros_G(const Gparameter& a, int scale)
699 int depth, trailing_zeros;
700 Gparameter::const_iterator last, dummyit;
701 last = check_parameter_G(a, scale, convergent, depth, trailing_zeros, dummyit);
703 GINAC_ASSERT(convergent);
705 if ((trailing_zeros > 0) && (depth > 0)) {
707 Gparameter new_a(a.begin(), a.end()-1);
708 result += G_eval1(0, scale) * trailing_zeros_G(new_a, scale);
709 for (Gparameter::const_iterator it = a.begin(); it != last; ++it) {
710 Gparameter new_a(a.begin(), it);
712 new_a.insert(new_a.end(), it, a.end()-1);
713 result -= trailing_zeros_G(new_a, scale);
716 return result / trailing_zeros;
718 return G_eval(a, scale);
723 // G transformation [VSW] (57),(58)
724 ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, int scale)
726 // pendint = ( y1, b1, ..., br )
727 // a = ( 0, ..., 0, amin )
730 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(0, ..., 0, sr; y2)
731 // where sr replaces amin
733 GINAC_ASSERT(a.back() != 0);
734 GINAC_ASSERT(a.size() > 0);
737 Gparameter new_pending_integrals = prepare_pending_integrals(pending_integrals, std::abs(a.back()));
738 const int psize = pending_integrals.size();
741 // G(sr_{+-}; y2 ) = G(y2_{-+}; sr) - G(0; sr) + ln(-y2_{-+})
746 result += log(gsyms[ex_to<numeric>(scale).to_int()]);
748 new_pending_integrals.push_back(-scale);
751 new_pending_integrals.push_back(scale);
755 result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals), pending_integrals.front());
759 result += trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals), new_pending_integrals.front());
762 new_pending_integrals.back() = 0;
763 result -= trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals), new_pending_integrals.front());
769 // G_m(sr_{+-}; y2) = -zeta_m + int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
770 // - int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
773 result -= zeta(a.size());
775 result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals), pending_integrals.front());
778 // term int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
779 // = int_0^sr dt/t G_{m-1}( t_{+-}; y2 )
780 Gparameter new_a(a.begin()+1, a.end());
781 new_pending_integrals.push_back(0);
782 result -= depth_one_trafo_G(new_pending_integrals, new_a, scale);
784 // term int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
785 // = int_0^y2 dt/t G_{m-1}( t_{+-}; y2 )
786 Gparameter new_pending_integrals_2;
787 new_pending_integrals_2.push_back(scale);
788 new_pending_integrals_2.push_back(0);
790 result += trailing_zeros_G(convert_pending_integrals_G(pending_integrals), pending_integrals.front())
791 * depth_one_trafo_G(new_pending_integrals_2, new_a, scale);
793 result += depth_one_trafo_G(new_pending_integrals_2, new_a, scale);
800 // forward declaration
801 ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
802 const Gparameter& pendint, const Gparameter& a_old, int scale);
805 // G transformation [VSW]
806 ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale)
808 // main recursion routine
810 // pendint = ( y1, b1, ..., br )
811 // a = ( a1, ..., amin, ..., aw )
814 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
815 // where sr replaces amin
817 // find smallest alpha, determine depth and trailing zeros, and check for convergence
819 int depth, trailing_zeros;
820 Gparameter::const_iterator min_it;
821 Gparameter::const_iterator firstzero =
822 check_parameter_G(a, scale, convergent, depth, trailing_zeros, min_it);
823 int min_it_pos = min_it - a.begin();
825 // special case: all a's are zero
832 result = G_eval(a, scale);
834 if (pendint.size() > 0) {
835 result *= trailing_zeros_G(convert_pending_integrals_G(pendint), pendint.front());
840 // handle trailing zeros
841 if (trailing_zeros > 0) {
843 Gparameter new_a(a.begin(), a.end()-1);
844 result += G_eval1(0, scale) * G_transform(pendint, new_a, scale);
845 for (Gparameter::const_iterator it = a.begin(); it != firstzero; ++it) {
846 Gparameter new_a(a.begin(), it);
848 new_a.insert(new_a.end(), it, a.end()-1);
849 result -= G_transform(pendint, new_a, scale);
851 return result / trailing_zeros;
856 if (pendint.size() > 0) {
857 return G_eval(convert_pending_integrals_G(pendint), pendint.front()) * G_eval(a, scale);
859 return G_eval(a, scale);
863 // call basic transformation for depth equal one
865 return depth_one_trafo_G(pendint, a, scale);
869 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
870 // = int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,0,...,aw,y2)
871 // + int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) int_0^{sr} ds_{r+1} d/ds_{r+1} G(a1,...,s_{r+1},...,aw,y2)
873 // smallest element in last place
874 if (min_it + 1 == a.end()) {
875 do { --min_it; } while (*min_it == 0);
877 Gparameter a1(a.begin(),min_it+1);
878 Gparameter a2(min_it+1,a.end());
880 ex result = G_transform(pendint,a2,scale)*G_transform(empty,a1,scale);
882 result -= shuffle_G(empty,a1,a2,pendint,a,scale);
887 Gparameter::iterator changeit;
889 // first term G(a_1,..,0,...,a_w;a_0)
890 Gparameter new_pendint = prepare_pending_integrals(pendint, a[min_it_pos]);
891 Gparameter new_a = a;
892 new_a[min_it_pos] = 0;
893 ex result = G_transform(empty, new_a, scale);
894 if (pendint.size() > 0) {
895 result *= trailing_zeros_G(convert_pending_integrals_G(pendint), pendint.front());
899 changeit = new_a.begin() + min_it_pos;
900 changeit = new_a.erase(changeit);
901 if (changeit != new_a.begin()) {
902 // smallest in the middle
903 new_pendint.push_back(*changeit);
904 result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
905 * G_transform(empty, new_a, scale);
906 int buffer = *changeit;
908 result += G_transform(new_pendint, new_a, scale);
910 new_pendint.pop_back();
912 new_pendint.push_back(*changeit);
913 result += trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
914 * G_transform(empty, new_a, scale);
916 result -= G_transform(new_pendint, new_a, scale);
918 // smallest at the front
919 new_pendint.push_back(scale);
920 result += trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
921 * G_transform(empty, new_a, scale);
922 new_pendint.back() = *changeit;
923 result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
924 * G_transform(empty, new_a, scale);
926 result += G_transform(new_pendint, new_a, scale);
932 // shuffles the two parameter list a1 and a2 and calls G_transform for every term except
933 // for the one that is equal to a_old
934 ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
935 const Gparameter& pendint, const Gparameter& a_old, int scale)
937 if (a1.size()==0 && a2.size()==0) {
938 // veto the one configuration we don't want
939 if ( a0 == a_old ) return 0;
941 return G_transform(pendint,a0,scale);
947 aa0.insert(aa0.end(),a1.begin(),a1.end());
948 return shuffle_G(aa0,empty,empty,pendint,a_old,scale);
954 aa0.insert(aa0.end(),a2.begin(),a2.end());
955 return shuffle_G(aa0,empty,empty,pendint,a_old,scale);
958 Gparameter a1_removed(a1.begin()+1,a1.end());
959 Gparameter a2_removed(a2.begin()+1,a2.end());
964 a01.push_back( a1[0] );
965 a02.push_back( a2[0] );
967 return shuffle_G(a01,a1_removed,a2,pendint,a_old,scale)
968 + shuffle_G(a02,a1,a2_removed,pendint,a_old,scale);
972 // handles the transformations and the numerical evaluation of G
973 // the parameter x, s and y must only contain numerics
974 ex G_numeric(const lst& x, const lst& s, const ex& y)
976 // check for convergence and necessary accelerations
977 bool need_trafo = false;
978 bool need_hoelder = false;
980 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
981 if (!(*it).is_zero()) {
983 if (abs(*it) - y < -pow(10,-Digits+1)) {
986 if (abs((abs(*it) - y)/y) < 0.01) {
991 if (x.op(x.nops()-1).is_zero()) {
994 if (depth == 1 && !need_trafo) {
995 return -Li(x.nops(), y / x.op(x.nops()-1)).evalf();
998 // do acceleration transformation (hoelder convolution [BBB])
1002 const int size = x.nops();
1004 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1005 newx.append(*it / y);
1008 for (int r=0; r<=size; ++r) {
1009 ex buffer = pow(-1, r);
1014 for (lst::const_iterator it = newx.begin(); it != newx.end(); ++it) {
1025 for (int j=r; j>=1; --j) {
1026 qlstx.append(1-newx.op(j-1));
1027 if (newx.op(j-1).info(info_flags::real) && newx.op(j-1) > 1 && newx.op(j-1) <= 2) {
1028 qlsts.append( s.op(j-1));
1030 qlsts.append( -s.op(j-1));
1033 if (qlstx.nops() > 0) {
1034 buffer *= G_numeric(qlstx, qlsts, 1/q);
1038 for (int j=r+1; j<=size; ++j) {
1039 plstx.append(newx.op(j-1));
1040 plsts.append(s.op(j-1));
1042 if (plstx.nops() > 0) {
1043 buffer *= G_numeric(plstx, plsts, 1/p);
1050 // convergence transformation
1053 // sort (|x|<->position) to determine indices
1054 std::multimap<ex,int> sortmap;
1056 for (int i=0; i<x.nops(); ++i) {
1057 if (!x[i].is_zero()) {
1058 sortmap.insert(std::pair<ex,int>(abs(x[i]), i));
1062 // include upper limit (scale)
1063 sortmap.insert(std::pair<ex,int>(abs(y), x.nops()));
1065 // generate missing dummy-symbols
1068 gsyms.push_back(symbol("GSYMS_ERROR"));
1070 for (std::multimap<ex,int>::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1071 if (it != sortmap.begin()) {
1072 if (it->second < x.nops()) {
1073 if (x[it->second] == lastentry) {
1074 gsyms.push_back(gsyms.back());
1078 if (y == lastentry) {
1079 gsyms.push_back(gsyms.back());
1084 std::ostringstream os;
1086 gsyms.push_back(symbol(os.str()));
1088 if (it->second < x.nops()) {
1089 lastentry = x[it->second];
1095 // fill position data according to sorted indices and prepare substitution list
1096 Gparameter a(x.nops());
1100 for (std::multimap<ex,int>::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1101 if (it->second < x.nops()) {
1102 if (s[it->second] > 0) {
1103 a[it->second] = pos;
1105 a[it->second] = -pos;
1107 subslst.append(gsyms[pos] == x[it->second]);
1110 subslst.append(gsyms[pos] == y);
1115 // do transformation
1117 ex result = G_transform(pendint, a, scale);
1118 // replace dummy symbols with their values
1119 result = result.eval().expand();
1120 result = result.subs(subslst).evalf();
1131 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1132 if ((*it).is_zero()) {
1135 newx.append(factor / (*it));
1143 return sign * numeric(mLi_do_summation(m, newx));
1147 ex mLi_numeric(const lst& m, const lst& x)
1149 // let G_numeric do the transformation
1153 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1154 for (int i = 1; i < *itm; ++i) {
1158 newx.append(factor / *itx);
1162 return pow(-1, m.nops()) * G_numeric(newx, s, _ex1);
1166 } // end of anonymous namespace
1169 //////////////////////////////////////////////////////////////////////
1171 // Generalized multiple polylogarithm G(x, y) and G(x, s, y)
1175 //////////////////////////////////////////////////////////////////////
1178 static ex G2_evalf(const ex& x_, const ex& y)
1180 if (!y.info(info_flags::positive)) {
1181 return G(x_, y).hold();
1183 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1184 if (x.nops() == 0) {
1188 return G(x_, y).hold();
1191 bool all_zero = true;
1192 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1193 if (!(*it).info(info_flags::numeric)) {
1194 return G(x_, y).hold();
1202 return pow(log(y), x.nops()) / factorial(x.nops());
1204 return G_numeric(x, s, y);
1208 static ex G2_eval(const ex& x_, const ex& y)
1210 //TODO eval to MZV or H or S or Lin
1212 if (!y.info(info_flags::positive)) {
1213 return G(x_, y).hold();
1215 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1216 if (x.nops() == 0) {
1220 return G(x_, y).hold();
1223 bool all_zero = true;
1224 bool crational = true;
1225 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1226 if (!(*it).info(info_flags::numeric)) {
1227 return G(x_, y).hold();
1229 if (!(*it).info(info_flags::crational)) {
1238 return pow(log(y), x.nops()) / factorial(x.nops());
1240 if (!y.info(info_flags::crational)) {
1244 return G(x_, y).hold();
1246 return G_numeric(x, s, y);
1250 unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2).
1251 evalf_func(G2_evalf).
1253 do_not_evalf_params().
1256 // derivative_func(G2_deriv).
1257 // print_func<print_latex>(G2_print_latex).
1260 static ex G3_evalf(const ex& x_, const ex& s_, const ex& y)
1262 if (!y.info(info_flags::positive)) {
1263 return G(x_, s_, y).hold();
1265 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1266 lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
1267 if (x.nops() != s.nops()) {
1268 return G(x_, s_, y).hold();
1270 if (x.nops() == 0) {
1274 return G(x_, s_, y).hold();
1277 bool all_zero = true;
1278 for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1279 if (!(*itx).info(info_flags::numeric)) {
1280 return G(x_, y).hold();
1282 if (!(*its).info(info_flags::real)) {
1283 return G(x_, y).hold();
1295 return pow(log(y), x.nops()) / factorial(x.nops());
1297 return G_numeric(x, sn, y);
1301 static ex G3_eval(const ex& x_, const ex& s_, const ex& y)
1303 //TODO eval to MZV or H or S or Lin
1305 if (!y.info(info_flags::positive)) {
1306 return G(x_, s_, y).hold();
1308 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1309 lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
1310 if (x.nops() != s.nops()) {
1311 return G(x_, s_, y).hold();
1313 if (x.nops() == 0) {
1317 return G(x_, s_, y).hold();
1320 bool all_zero = true;
1321 bool crational = true;
1322 for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1323 if (!(*itx).info(info_flags::numeric)) {
1324 return G(x_, s_, y).hold();
1326 if (!(*its).info(info_flags::real)) {
1327 return G(x_, s_, y).hold();
1329 if (!(*itx).info(info_flags::crational)) {
1342 return pow(log(y), x.nops()) / factorial(x.nops());
1344 if (!y.info(info_flags::crational)) {
1348 return G(x_, s_, y).hold();
1350 return G_numeric(x, sn, y);
1354 unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3).
1355 evalf_func(G3_evalf).
1357 do_not_evalf_params().
1360 // derivative_func(G3_deriv).
1361 // print_func<print_latex>(G3_print_latex).
1364 //////////////////////////////////////////////////////////////////////
1366 // Classical polylogarithm and multiple polylogarithm Li(m,x)
1370 //////////////////////////////////////////////////////////////////////
1373 static ex Li_evalf(const ex& m_, const ex& x_)
1375 // classical polylogs
1376 if (m_.info(info_flags::posint)) {
1377 if (x_.info(info_flags::numeric)) {
1378 return Lin_numeric(ex_to<numeric>(m_).to_int(), ex_to<numeric>(x_));
1380 // try to numerically evaluate second argument
1381 ex x_val = x_.evalf();
1382 if (x_val.info(info_flags::numeric)) {
1383 return Lin_numeric(ex_to<numeric>(m_).to_int(), ex_to<numeric>(x_val));
1387 // multiple polylogs
1388 if (is_a<lst>(m_) && is_a<lst>(x_)) {
1390 const lst& m = ex_to<lst>(m_);
1391 const lst& x = ex_to<lst>(x_);
1392 if (m.nops() != x.nops()) {
1393 return Li(m_,x_).hold();
1395 if (x.nops() == 0) {
1398 if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) {
1399 return Li(m_,x_).hold();
1402 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1403 if (!(*itm).info(info_flags::posint)) {
1404 return Li(m_, x_).hold();
1406 if (!(*itx).info(info_flags::numeric)) {
1407 return Li(m_, x_).hold();
1414 return mLi_numeric(m, x);
1417 return Li(m_,x_).hold();
1421 static ex Li_eval(const ex& m_, const ex& x_)
1423 if (is_a<lst>(m_)) {
1424 if (is_a<lst>(x_)) {
1425 // multiple polylogs
1426 const lst& m = ex_to<lst>(m_);
1427 const lst& x = ex_to<lst>(x_);
1428 if (m.nops() != x.nops()) {
1429 return Li(m_,x_).hold();
1431 if (x.nops() == 0) {
1435 bool is_zeta = true;
1436 bool do_evalf = true;
1437 bool crational = true;
1438 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1439 if (!(*itm).info(info_flags::posint)) {
1440 return Li(m_,x_).hold();
1442 if ((*itx != _ex1) && (*itx != _ex_1)) {
1443 if (itx != x.begin()) {
1451 if (!(*itx).info(info_flags::numeric)) {
1454 if (!(*itx).info(info_flags::crational)) {
1463 lst newm = convert_parameter_Li_to_H(m, x, prefactor);
1464 return prefactor * H(newm, x[0]);
1466 if (do_evalf && !crational) {
1467 return mLi_numeric(m,x);
1470 return Li(m_, x_).hold();
1471 } else if (is_a<lst>(x_)) {
1472 return Li(m_, x_).hold();
1475 // classical polylogs
1483 return (pow(2,1-m_)-1) * zeta(m_);
1489 if (x_.is_equal(I)) {
1490 return power(Pi,_ex2)/_ex_48 + Catalan*I;
1492 if (x_.is_equal(-I)) {
1493 return power(Pi,_ex2)/_ex_48 - Catalan*I;
1496 if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
1497 return Lin_numeric(ex_to<numeric>(m_).to_int(), ex_to<numeric>(x_));
1500 return Li(m_, x_).hold();
1504 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
1506 if (is_a<lst>(m) || is_a<lst>(x)) {
1509 seq.push_back(expair(Li(m, x), 0));
1510 return pseries(rel, seq);
1513 // classical polylog
1514 const ex x_pt = x.subs(rel, subs_options::no_pattern);
1515 if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
1516 // First special case: x==0 (derivatives have poles)
1517 if (x_pt.is_zero()) {
1520 // manually construct the primitive expansion
1521 for (int i=1; i<order; ++i)
1522 ser += pow(s,i) / pow(numeric(i), m);
1523 // substitute the argument's series expansion
1524 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
1525 // maybe that was terminating, so add a proper order term
1527 nseq.push_back(expair(Order(_ex1), order));
1528 ser += pseries(rel, nseq);
1529 // reexpanding it will collapse the series again
1530 return ser.series(rel, order);
1532 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
1533 throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
1535 // all other cases should be safe, by now:
1536 throw do_taylor(); // caught by function::series()
1540 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
1542 GINAC_ASSERT(deriv_param < 2);
1543 if (deriv_param == 0) {
1546 if (m_.nops() > 1) {
1547 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
1550 if (is_a<lst>(m_)) {
1556 if (is_a<lst>(x_)) {
1562 return Li(m-1, x) / x;
1569 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
1572 if (is_a<lst>(m_)) {
1578 if (is_a<lst>(x_)) {
1583 c.s << "\\mbox{Li}_{";
1584 lst::const_iterator itm = m.begin();
1587 for (; itm != m.end(); itm++) {
1592 lst::const_iterator itx = x.begin();
1595 for (; itx != x.end(); itx++) {
1603 REGISTER_FUNCTION(Li,
1604 evalf_func(Li_evalf).
1606 series_func(Li_series).
1607 derivative_func(Li_deriv).
1608 print_func<print_latex>(Li_print_latex).
1609 do_not_evalf_params());
1612 //////////////////////////////////////////////////////////////////////
1614 // Nielsen's generalized polylogarithm S(n,p,x)
1618 //////////////////////////////////////////////////////////////////////
1621 // anonymous namespace for helper functions
1625 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
1627 std::vector<std::vector<cln::cl_N> > Yn;
1628 int ynsize = 0; // number of Yn[]
1629 int ynlength = 100; // initial length of all Yn[i]
1632 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
1633 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
1634 // representing S_{n,p}(x).
1635 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
1636 // equivalent Z-sum.
1637 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
1638 // representing S_{n,p}(x).
1639 // The calculation of Y_n uses the values from Y_{n-1}.
1640 void fill_Yn(int n, const cln::float_format_t& prec)
1642 const int initsize = ynlength;
1643 //const int initsize = initsize_Yn;
1644 cln::cl_N one = cln::cl_float(1, prec);
1647 std::vector<cln::cl_N> buf(initsize);
1648 std::vector<cln::cl_N>::iterator it = buf.begin();
1649 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1650 *it = (*itprev) / cln::cl_N(n+1) * one;
1653 // sums with an index smaller than the depth are zero and need not to be calculated.
1654 // calculation starts with depth, which is n+2)
1655 for (int i=n+2; i<=initsize+n; i++) {
1656 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1662 std::vector<cln::cl_N> buf(initsize);
1663 std::vector<cln::cl_N>::iterator it = buf.begin();
1666 for (int i=2; i<=initsize; i++) {
1667 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1676 // make Yn longer ...
1677 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
1680 cln::cl_N one = cln::cl_float(1, prec);
1682 Yn[0].resize(newsize);
1683 std::vector<cln::cl_N>::iterator it = Yn[0].begin();
1685 for (int i=ynlength+1; i<=newsize; i++) {
1686 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1690 for (int n=1; n<ynsize; n++) {
1691 Yn[n].resize(newsize);
1692 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
1693 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1696 for (int i=ynlength+n+1; i<=newsize+n; i++) {
1697 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1707 // helper function for S(n,p,x)
1709 cln::cl_N C(int n, int p)
1713 for (int k=0; k<p; k++) {
1714 for (int j=0; j<=(n+k-1)/2; j++) {
1718 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
1721 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
1728 result = result + cln::factorial(n+k-1)
1729 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
1730 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1733 result = result - cln::factorial(n+k-1)
1734 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
1735 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1740 result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
1741 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1744 result = result + cln::factorial(n+k-1)
1745 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
1746 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1754 if (((np)/2+n) & 1) {
1755 result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1758 result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1766 // helper function for S(n,p,x)
1767 // [Kol] remark to (9.1)
1768 cln::cl_N a_k(int k)
1777 for (int m=2; m<=k; m++) {
1778 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
1785 // helper function for S(n,p,x)
1786 // [Kol] remark to (9.1)
1787 cln::cl_N b_k(int k)
1796 for (int m=2; m<=k; m++) {
1797 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
1804 // helper function for S(n,p,x)
1805 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1808 return Li_projection(n+1, x, prec);
1811 // check if precalculated values are sufficient
1813 for (int i=ynsize; i<p-1; i++) {
1818 // should be done otherwise
1819 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1820 cln::cl_N xf = x * one;
1821 //cln::cl_N xf = x * cln::cl_float(1, prec);
1825 cln::cl_N factor = cln::expt(xf, p);
1829 if (i-p >= ynlength) {
1831 make_Yn_longer(ynlength*2, prec);
1833 res = res + factor / cln::expt(cln::cl_I(i),n+1) * Yn[p-2][i-p]; // should we check it? or rely on magic number? ...
1834 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
1835 factor = factor * xf;
1837 } while (res != resbuf);
1843 // helper function for S(n,p,x)
1844 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1847 if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
1849 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
1850 * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
1852 for (int s=0; s<n; s++) {
1854 for (int r=0; r<p; r++) {
1855 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
1856 * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
1858 result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
1864 return S_do_sum(n, p, x, prec);
1868 // helper function for S(n,p,x)
1869 numeric S_num(int n, int p, const numeric& x)
1873 // [Kol] (2.22) with (2.21)
1874 return cln::zeta(p+1);
1879 return cln::zeta(n+1);
1884 for (int nu=0; nu<n; nu++) {
1885 for (int rho=0; rho<=p; rho++) {
1886 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
1887 * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
1890 result = result * cln::expt(cln::cl_I(-1),n+p-1);
1897 return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
1899 // throw std::runtime_error("don't know how to evaluate this function!");
1902 // what is the desired float format?
1903 // first guess: default format
1904 cln::float_format_t prec = cln::default_float_format;
1905 const cln::cl_N value = x.to_cl_N();
1906 // second guess: the argument's format
1907 if (!x.real().is_rational())
1908 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
1909 else if (!x.imag().is_rational())
1910 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
1913 if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95))) {
1915 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
1916 * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
1918 for (int s=0; s<n; s++) {
1920 for (int r=0; r<p; r++) {
1921 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
1922 * S_num(p-r,n-s,1-value).to_cl_N() / cln::factorial(r);
1924 result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
1931 if (cln::abs(value) > 1) {
1935 for (int s=0; s<p; s++) {
1936 for (int r=0; r<=s; r++) {
1937 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
1938 / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
1939 * S_num(n+s-r,p-s,cln::recip(value)).to_cl_N();
1942 result = result * cln::expt(cln::cl_I(-1),n);
1945 for (int r=0; r<n; r++) {
1946 res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
1948 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
1950 result = result + cln::expt(cln::cl_I(-1),p) * res2;
1955 return S_projection(n, p, value, prec);
1960 } // end of anonymous namespace
1963 //////////////////////////////////////////////////////////////////////
1965 // Nielsen's generalized polylogarithm S(n,p,x)
1969 //////////////////////////////////////////////////////////////////////
1972 static ex S_evalf(const ex& n, const ex& p, const ex& x)
1974 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
1975 if (is_a<numeric>(x)) {
1976 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1978 ex x_val = x.evalf();
1979 if (is_a<numeric>(x_val)) {
1980 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x_val));
1984 return S(n, p, x).hold();
1988 static ex S_eval(const ex& n, const ex& p, const ex& x)
1990 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
1996 for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
2004 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2005 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
2010 return pow(-log(1-x), p) / factorial(p);
2012 return S(n, p, x).hold();
2016 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
2019 return Li(n+1, x).series(rel, order, options);
2022 const ex x_pt = x.subs(rel, subs_options::no_pattern);
2023 if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
2024 // First special case: x==0 (derivatives have poles)
2025 if (x_pt.is_zero()) {
2028 // manually construct the primitive expansion
2029 // subsum = Euler-Zagier-Sum is needed
2030 // dirty hack (slow ...) calculation of subsum:
2031 std::vector<ex> presubsum, subsum;
2032 subsum.push_back(0);
2033 for (int i=1; i<order-1; ++i) {
2034 subsum.push_back(subsum[i-1] + numeric(1, i));
2036 for (int depth=2; depth<p; ++depth) {
2038 for (int i=1; i<order-1; ++i) {
2039 subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
2043 for (int i=1; i<order; ++i) {
2044 ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
2046 // substitute the argument's series expansion
2047 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
2048 // maybe that was terminating, so add a proper order term
2050 nseq.push_back(expair(Order(_ex1), order));
2051 ser += pseries(rel, nseq);
2052 // reexpanding it will collapse the series again
2053 return ser.series(rel, order);
2055 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
2056 throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
2058 // all other cases should be safe, by now:
2059 throw do_taylor(); // caught by function::series()
2063 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
2065 GINAC_ASSERT(deriv_param < 3);
2066 if (deriv_param < 2) {
2070 return S(n-1, p, x) / x;
2072 return S(n, p-1, x) / (1-x);
2077 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
2079 c.s << "\\mbox{S}_{";
2089 REGISTER_FUNCTION(S,
2090 evalf_func(S_evalf).
2092 series_func(S_series).
2093 derivative_func(S_deriv).
2094 print_func<print_latex>(S_print_latex).
2095 do_not_evalf_params());
2098 //////////////////////////////////////////////////////////////////////
2100 // Harmonic polylogarithm H(m,x)
2104 //////////////////////////////////////////////////////////////////////
2107 // anonymous namespace for helper functions
2111 // regulates the pole (used by 1/x-transformation)
2112 symbol H_polesign("IMSIGN");
2115 // convert parameters from H to Li representation
2116 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
2117 // returns true if some parameters are negative
2118 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
2120 // expand parameter list
2122 for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
2124 for (ex count=*it-1; count > 0; count--) {
2128 } else if (*it < -1) {
2129 for (ex count=*it+1; count < 0; count++) {
2140 bool has_negative_parameters = false;
2142 for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
2148 m.append((*it+acc-1) * signum);
2150 m.append((*it-acc+1) * signum);
2156 has_negative_parameters = true;
2159 if (has_negative_parameters) {
2160 for (int i=0; i<m.nops(); i++) {
2162 m.let_op(i) = -m.op(i);
2170 return has_negative_parameters;
2174 // recursivly transforms H to corresponding multiple polylogarithms
2175 struct map_trafo_H_convert_to_Li : public map_function
2177 ex operator()(const ex& e)
2179 if (is_a<add>(e) || is_a<mul>(e)) {
2180 return e.map(*this);
2182 if (is_a<function>(e)) {
2183 std::string name = ex_to<function>(e).get_name();
2186 if (is_a<lst>(e.op(0))) {
2187 parameter = ex_to<lst>(e.op(0));
2189 parameter = lst(e.op(0));
2196 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2197 s.let_op(0) = s.op(0) * arg;
2198 return pf * Li(m, s).hold();
2200 for (int i=0; i<m.nops(); i++) {
2203 s.let_op(0) = s.op(0) * arg;
2204 return Li(m, s).hold();
2213 // recursivly transforms H to corresponding zetas
2214 struct map_trafo_H_convert_to_zeta : public map_function
2216 ex operator()(const ex& e)
2218 if (is_a<add>(e) || is_a<mul>(e)) {
2219 return e.map(*this);
2221 if (is_a<function>(e)) {
2222 std::string name = ex_to<function>(e).get_name();
2225 if (is_a<lst>(e.op(0))) {
2226 parameter = ex_to<lst>(e.op(0));
2228 parameter = lst(e.op(0));
2234 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2235 return pf * zeta(m, s);
2246 // remove trailing zeros from H-parameters
2247 struct map_trafo_H_reduce_trailing_zeros : public map_function
2249 ex operator()(const ex& e)
2251 if (is_a<add>(e) || is_a<mul>(e)) {
2252 return e.map(*this);
2254 if (is_a<function>(e)) {
2255 std::string name = ex_to<function>(e).get_name();
2258 if (is_a<lst>(e.op(0))) {
2259 parameter = ex_to<lst>(e.op(0));
2261 parameter = lst(e.op(0));
2264 if (parameter.op(parameter.nops()-1) == 0) {
2267 if (parameter.nops() == 1) {
2272 lst::const_iterator it = parameter.begin();
2273 while ((it != parameter.end()) && (*it == 0)) {
2276 if (it == parameter.end()) {
2277 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
2281 parameter.remove_last();
2282 int lastentry = parameter.nops();
2283 while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
2288 ex result = log(arg) * H(parameter,arg).hold();
2290 for (ex i=0; i<lastentry; i++) {
2291 if (parameter[i] > 0) {
2293 result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
2296 } else if (parameter[i] < 0) {
2298 result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
2306 if (lastentry < parameter.nops()) {
2307 result = result / (parameter.nops()-lastentry+1);
2308 return result.map(*this);
2320 // returns an expression with zeta functions corresponding to the parameter list for H
2321 ex convert_H_to_zeta(const lst& m)
2323 symbol xtemp("xtemp");
2324 map_trafo_H_reduce_trailing_zeros filter;
2325 map_trafo_H_convert_to_zeta filter2;
2326 return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
2330 // convert signs form Li to H representation
2331 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
2334 lst::const_iterator itm = m.begin();
2335 lst::const_iterator itx = ++x.begin();
2340 while (itx != x.end()) {
2341 signum *= (*itx > 0) ? 1 : -1;
2343 res.append((*itm) * signum);
2351 // multiplies an one-dimensional H with another H
2353 ex trafo_H_mult(const ex& h1, const ex& h2)
2358 ex h1nops = h1.op(0).nops();
2359 ex h2nops = h2.op(0).nops();
2361 hshort = h2.op(0).op(0);
2362 hlong = ex_to<lst>(h1.op(0));
2364 hshort = h1.op(0).op(0);
2366 hlong = ex_to<lst>(h2.op(0));
2368 hlong = h2.op(0).op(0);
2371 for (int i=0; i<=hlong.nops(); i++) {
2375 newparameter.append(hlong[j]);
2377 newparameter.append(hshort);
2378 for (; j<hlong.nops(); j++) {
2379 newparameter.append(hlong[j]);
2381 res += H(newparameter, h1.op(1)).hold();
2387 // applies trafo_H_mult recursively on expressions
2388 struct map_trafo_H_mult : public map_function
2390 ex operator()(const ex& e)
2393 return e.map(*this);
2401 for (int pos=0; pos<e.nops(); pos++) {
2402 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
2403 std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
2405 for (ex i=0; i<e.op(pos).op(1); i++) {
2406 Hlst.append(e.op(pos).op(0));
2410 } else if (is_a<function>(e.op(pos))) {
2411 std::string name = ex_to<function>(e.op(pos)).get_name();
2413 if (e.op(pos).op(0).nops() > 1) {
2416 Hlst.append(e.op(pos));
2421 result *= e.op(pos);
2424 if (Hlst.nops() > 0) {
2425 firstH = Hlst[Hlst.nops()-1];
2432 if (Hlst.nops() > 0) {
2433 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
2435 for (int i=1; i<Hlst.nops(); i++) {
2436 result *= Hlst.op(i);
2438 result = result.expand();
2439 map_trafo_H_mult recursion;
2440 return recursion(result);
2451 // do integration [ReV] (55)
2452 // put parameter 0 in front of existing parameters
2453 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
2457 if (is_a<function>(e)) {
2458 name = ex_to<function>(e).get_name();
2463 for (int i=0; i<e.nops(); i++) {
2464 if (is_a<function>(e.op(i))) {
2465 std::string name = ex_to<function>(e.op(i)).get_name();
2473 lst newparameter = ex_to<lst>(h.op(0));
2474 newparameter.prepend(0);
2475 ex addzeta = convert_H_to_zeta(newparameter);
2476 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2478 return e * (-H(lst(0),1/arg).hold());
2483 // do integration [ReV] (49)
2484 // put parameter 1 in front of existing parameters
2485 ex trafo_H_prepend_one(const ex& e, const ex& arg)
2489 if (is_a<function>(e)) {
2490 name = ex_to<function>(e).get_name();
2495 for (int i=0; i<e.nops(); i++) {
2496 if (is_a<function>(e.op(i))) {
2497 std::string name = ex_to<function>(e.op(i)).get_name();
2505 lst newparameter = ex_to<lst>(h.op(0));
2506 newparameter.prepend(1);
2507 return e.subs(h == H(newparameter, h.op(1)).hold());
2509 return e * H(lst(1),1-arg).hold();
2514 // do integration [ReV] (55)
2515 // put parameter -1 in front of existing parameters
2516 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
2520 if (is_a<function>(e)) {
2521 name = ex_to<function>(e).get_name();
2526 for (int i=0; i<e.nops(); i++) {
2527 if (is_a<function>(e.op(i))) {
2528 std::string name = ex_to<function>(e.op(i)).get_name();
2536 lst newparameter = ex_to<lst>(h.op(0));
2537 newparameter.prepend(-1);
2538 ex addzeta = convert_H_to_zeta(newparameter);
2539 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2541 ex addzeta = convert_H_to_zeta(lst(-1));
2542 return (e * (addzeta - H(lst(-1),1/arg).hold())).expand();
2547 // do integration [ReV] (55)
2548 // put parameter -1 in front of existing parameters
2549 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
2553 if (is_a<function>(e)) {
2554 name = ex_to<function>(e).get_name();
2559 for (int i=0; i<e.nops(); i++) {
2560 if (is_a<function>(e.op(i))) {
2561 std::string name = ex_to<function>(e.op(i)).get_name();
2569 lst newparameter = ex_to<lst>(h.op(0));
2570 newparameter.prepend(-1);
2571 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2573 return (e * H(lst(-1),(1-arg)/(1+arg)).hold()).expand();
2578 // do integration [ReV] (55)
2579 // put parameter 1 in front of existing parameters
2580 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
2584 if (is_a<function>(e)) {
2585 name = ex_to<function>(e).get_name();
2590 for (int i=0; i<e.nops(); i++) {
2591 if (is_a<function>(e.op(i))) {
2592 std::string name = ex_to<function>(e.op(i)).get_name();
2600 lst newparameter = ex_to<lst>(h.op(0));
2601 newparameter.prepend(1);
2602 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2604 return (e * H(lst(1),(1-arg)/(1+arg)).hold()).expand();
2609 // do x -> 1-x transformation
2610 struct map_trafo_H_1mx : public map_function
2612 ex operator()(const ex& e)
2614 if (is_a<add>(e) || is_a<mul>(e)) {
2615 return e.map(*this);
2618 if (is_a<function>(e)) {
2619 std::string name = ex_to<function>(e).get_name();
2622 lst parameter = ex_to<lst>(e.op(0));
2625 // special cases if all parameters are either 0, 1 or -1
2626 bool allthesame = true;
2627 if (parameter.op(0) == 0) {
2628 for (int i=1; i<parameter.nops(); i++) {
2629 if (parameter.op(i) != 0) {
2636 for (int i=parameter.nops(); i>0; i--) {
2637 newparameter.append(1);
2639 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2641 } else if (parameter.op(0) == -1) {
2642 throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
2644 for (int i=1; i<parameter.nops(); i++) {
2645 if (parameter.op(i) != 1) {
2652 for (int i=parameter.nops(); i>0; i--) {
2653 newparameter.append(0);
2655 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2659 lst newparameter = parameter;
2660 newparameter.remove_first();
2662 if (parameter.op(0) == 0) {
2665 ex res = convert_H_to_zeta(parameter);
2666 //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
2667 map_trafo_H_1mx recursion;
2668 ex buffer = recursion(H(newparameter, arg).hold());
2669 if (is_a<add>(buffer)) {
2670 for (int i=0; i<buffer.nops(); i++) {
2671 res -= trafo_H_prepend_one(buffer.op(i), arg);
2674 res -= trafo_H_prepend_one(buffer, arg);
2681 map_trafo_H_1mx recursion;
2682 map_trafo_H_mult unify;
2683 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
2685 while (parameter.op(firstzero) == 1) {
2688 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
2692 newparameter.append(parameter[j+1]);
2694 newparameter.append(1);
2695 for (; j<parameter.nops()-1; j++) {
2696 newparameter.append(parameter[j+1]);
2698 res -= H(newparameter, arg).hold();
2700 res = recursion(res).expand() / firstzero;
2710 // do x -> 1/x transformation
2711 struct map_trafo_H_1overx : public map_function
2713 ex operator()(const ex& e)
2715 if (is_a<add>(e) || is_a<mul>(e)) {
2716 return e.map(*this);
2719 if (is_a<function>(e)) {
2720 std::string name = ex_to<function>(e).get_name();
2723 lst parameter = ex_to<lst>(e.op(0));
2726 // special cases if all parameters are either 0, 1 or -1
2727 bool allthesame = true;
2728 if (parameter.op(0) == 0) {
2729 for (int i=1; i<parameter.nops(); i++) {
2730 if (parameter.op(i) != 0) {
2736 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
2738 } else if (parameter.op(0) == -1) {
2739 for (int i=1; i<parameter.nops(); i++) {
2740 if (parameter.op(i) != -1) {
2746 map_trafo_H_mult unify;
2747 return unify((pow(H(lst(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops())
2748 / factorial(parameter.nops())).expand());
2751 for (int i=1; i<parameter.nops(); i++) {
2752 if (parameter.op(i) != 1) {
2758 map_trafo_H_mult unify;
2759 return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() + H_polesign, parameter.nops())
2760 / factorial(parameter.nops())).expand());
2764 lst newparameter = parameter;
2765 newparameter.remove_first();
2767 if (parameter.op(0) == 0) {
2770 ex res = convert_H_to_zeta(parameter);
2771 map_trafo_H_1overx recursion;
2772 ex buffer = recursion(H(newparameter, arg).hold());
2773 if (is_a<add>(buffer)) {
2774 for (int i=0; i<buffer.nops(); i++) {
2775 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
2778 res += trafo_H_1tx_prepend_zero(buffer, arg);
2782 } else if (parameter.op(0) == -1) {
2784 // leading negative one
2785 ex res = convert_H_to_zeta(parameter);
2786 map_trafo_H_1overx recursion;
2787 ex buffer = recursion(H(newparameter, arg).hold());
2788 if (is_a<add>(buffer)) {
2789 for (int i=0; i<buffer.nops(); i++) {
2790 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
2793 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
2800 map_trafo_H_1overx recursion;
2801 map_trafo_H_mult unify;
2802 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
2804 while (parameter.op(firstzero) == 1) {
2807 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
2811 newparameter.append(parameter[j+1]);
2813 newparameter.append(1);
2814 for (; j<parameter.nops()-1; j++) {
2815 newparameter.append(parameter[j+1]);
2817 res -= H(newparameter, arg).hold();
2819 res = recursion(res).expand() / firstzero;
2831 // do x -> (1-x)/(1+x) transformation
2832 struct map_trafo_H_1mxt1px : public map_function
2834 ex operator()(const ex& e)
2836 if (is_a<add>(e) || is_a<mul>(e)) {
2837 return e.map(*this);
2840 if (is_a<function>(e)) {
2841 std::string name = ex_to<function>(e).get_name();
2844 lst parameter = ex_to<lst>(e.op(0));
2847 // special cases if all parameters are either 0, 1 or -1
2848 bool allthesame = true;
2849 if (parameter.op(0) == 0) {
2850 for (int i=1; i<parameter.nops(); i++) {
2851 if (parameter.op(i) != 0) {
2857 map_trafo_H_mult unify;
2858 return unify((pow(-H(lst(1),(1-arg)/(1+arg)).hold() - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
2859 / factorial(parameter.nops())).expand());
2861 } else if (parameter.op(0) == -1) {
2862 for (int i=1; i<parameter.nops(); i++) {
2863 if (parameter.op(i) != -1) {
2869 map_trafo_H_mult unify;
2870 return unify((pow(log(2) - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
2871 / factorial(parameter.nops())).expand());
2874 for (int i=1; i<parameter.nops(); i++) {
2875 if (parameter.op(i) != 1) {
2881 map_trafo_H_mult unify;
2882 return unify((pow(-log(2) - H(lst(0),(1-arg)/(1+arg)).hold() + H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
2883 / factorial(parameter.nops())).expand());
2887 lst newparameter = parameter;
2888 newparameter.remove_first();
2890 if (parameter.op(0) == 0) {
2893 ex res = convert_H_to_zeta(parameter);
2894 map_trafo_H_1mxt1px recursion;
2895 ex buffer = recursion(H(newparameter, arg).hold());
2896 if (is_a<add>(buffer)) {
2897 for (int i=0; i<buffer.nops(); i++) {
2898 res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
2901 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
2905 } else if (parameter.op(0) == -1) {
2907 // leading negative one
2908 ex res = convert_H_to_zeta(parameter);
2909 map_trafo_H_1mxt1px recursion;
2910 ex buffer = recursion(H(newparameter, arg).hold());
2911 if (is_a<add>(buffer)) {
2912 for (int i=0; i<buffer.nops(); i++) {
2913 res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
2916 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
2923 map_trafo_H_1mxt1px recursion;
2924 map_trafo_H_mult unify;
2925 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
2927 while (parameter.op(firstzero) == 1) {
2930 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
2934 newparameter.append(parameter[j+1]);
2936 newparameter.append(1);
2937 for (; j<parameter.nops()-1; j++) {
2938 newparameter.append(parameter[j+1]);
2940 res -= H(newparameter, arg).hold();
2942 res = recursion(res).expand() / firstzero;
2954 // do the actual summation.
2955 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
2957 const int j = m.size();
2959 std::vector<cln::cl_N> t(j);
2961 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2962 cln::cl_N factor = cln::expt(x, j) * one;
2968 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
2969 for (int k=j-2; k>=1; k--) {
2970 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
2972 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
2973 factor = factor * x;
2974 } while (t[0] != t0buf);
2980 } // end of anonymous namespace
2983 //////////////////////////////////////////////////////////////////////
2985 // Harmonic polylogarithm H(m,x)
2989 //////////////////////////////////////////////////////////////////////
2992 static ex H_evalf(const ex& x1, const ex& x2)
2994 if (is_a<lst>(x1)) {
2997 if (is_a<numeric>(x2)) {
2998 x = ex_to<numeric>(x2).to_cl_N();
3000 ex x2_val = x2.evalf();
3001 if (is_a<numeric>(x2_val)) {
3002 x = ex_to<numeric>(x2_val).to_cl_N();
3006 for (int i=0; i<x1.nops(); i++) {
3007 if (!x1.op(i).info(info_flags::integer)) {
3008 return H(x1, x2).hold();
3011 if (x1.nops() < 1) {
3012 return H(x1, x2).hold();
3015 const lst& morg = ex_to<lst>(x1);
3016 // remove trailing zeros ...
3017 if (*(--morg.end()) == 0) {
3018 symbol xtemp("xtemp");
3019 map_trafo_H_reduce_trailing_zeros filter;
3020 return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
3022 // ... and expand parameter notation
3023 bool has_minus_one = false;
3025 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
3027 for (ex count=*it-1; count > 0; count--) {
3031 } else if (*it <= -1) {
3032 for (ex count=*it+1; count < 0; count++) {
3036 has_minus_one = true;
3043 if (cln::abs(x) < 0.95) {
3047 if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
3048 // negative parameters -> s_lst is filled
3049 std::vector<int> m_int;
3050 std::vector<cln::cl_N> x_cln;
3051 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
3052 it_int != m_lst.end(); it_int++, it_cln++) {
3053 m_int.push_back(ex_to<numeric>(*it_int).to_int());
3054 x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
3056 x_cln.front() = x_cln.front() * x;
3057 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
3059 // only positive parameters
3061 if (m_lst.nops() == 1) {
3062 return Li(m_lst.op(0), x2).evalf();
3064 std::vector<int> m_int;
3065 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
3066 m_int.push_back(ex_to<numeric>(*it).to_int());
3068 return numeric(H_do_sum(m_int, x));
3072 symbol xtemp("xtemp");
3075 // ensure that the realpart of the argument is positive
3076 if (cln::realpart(x) < 0) {
3078 for (int i=0; i<m.nops(); i++) {
3080 m.let_op(i) = -m.op(i);
3087 if (cln::abs(x) >= 2.0) {
3088 map_trafo_H_1overx trafo;
3089 res *= trafo(H(m, xtemp));
3090 if (cln::imagpart(x) <= 0) {
3091 res = res.subs(H_polesign == -I*Pi);
3093 res = res.subs(H_polesign == I*Pi);
3095 return res.subs(xtemp == numeric(x)).evalf();
3098 // check transformations for 0.95 <= |x| < 2.0
3100 // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47
3101 if (cln::abs(x-9.53) <= 9.47) {
3103 map_trafo_H_1mxt1px trafo;
3104 res *= trafo(H(m, xtemp));
3107 if (has_minus_one) {
3108 map_trafo_H_convert_to_Li filter;
3109 return filter(H(m, numeric(x)).hold()).evalf();
3111 map_trafo_H_1mx trafo;
3112 res *= trafo(H(m, xtemp));
3115 return res.subs(xtemp == numeric(x)).evalf();
3118 return H(x1,x2).hold();
3122 static ex H_eval(const ex& m_, const ex& x)
3125 if (is_a<lst>(m_)) {
3130 if (m.nops() == 0) {
3138 if (*m.begin() > _ex1) {
3144 } else if (*m.begin() < _ex_1) {
3150 } else if (*m.begin() == _ex0) {
3157 for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
3158 if ((*it).info(info_flags::integer)) {
3169 } else if (*it < _ex_1) {
3189 } else if (step == 1) {
3201 // if some m_i is not an integer
3202 return H(m_, x).hold();
3205 if ((x == _ex1) && (*(--m.end()) != _ex0)) {
3206 return convert_H_to_zeta(m);
3212 return H(m_, x).hold();
3214 return pow(log(x), m.nops()) / factorial(m.nops());
3217 return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
3219 } else if ((step == 1) && (pos1 == _ex0)){
3224 return pow(-1, p) * S(n, p, -x);
3230 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
3231 return H(m_, x).evalf();
3233 return H(m_, x).hold();
3237 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
3240 seq.push_back(expair(H(m, x), 0));
3241 return pseries(rel, seq);
3245 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
3247 GINAC_ASSERT(deriv_param < 2);
3248 if (deriv_param == 0) {
3252 if (is_a<lst>(m_)) {
3268 return 1/(1-x) * H(m, x);
3269 } else if (mb == _ex_1) {
3270 return 1/(1+x) * H(m, x);
3277 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
3280 if (is_a<lst>(m_)) {
3285 c.s << "\\mbox{H}_{";
3286 lst::const_iterator itm = m.begin();
3289 for (; itm != m.end(); itm++) {
3299 REGISTER_FUNCTION(H,
3300 evalf_func(H_evalf).
3302 series_func(H_series).
3303 derivative_func(H_deriv).
3304 print_func<print_latex>(H_print_latex).
3305 do_not_evalf_params());
3308 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
3309 ex convert_H_to_Li(const ex& m, const ex& x)
3311 map_trafo_H_reduce_trailing_zeros filter;
3312 map_trafo_H_convert_to_Li filter2;
3314 return filter2(filter(H(m, x).hold()));
3316 return filter2(filter(H(lst(m), x).hold()));
3321 //////////////////////////////////////////////////////////////////////
3323 // Multiple zeta values zeta(x) and zeta(x,s)
3327 //////////////////////////////////////////////////////////////////////
3330 // anonymous namespace for helper functions
3334 // parameters and data for [Cra] algorithm
3335 const cln::cl_N lambda = cln::cl_N("319/320");
3338 std::vector<std::vector<cln::cl_N> > f_kj;
3339 std::vector<cln::cl_N> crB;
3340 std::vector<std::vector<cln::cl_N> > crG;
3341 std::vector<cln::cl_N> crX;
3344 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
3346 const int size = a.size();
3347 for (int n=0; n<size; n++) {
3349 for (int m=0; m<=n; m++) {
3350 c[n] = c[n] + a[m]*b[n-m];
3357 void initcX(const std::vector<int>& s)
3359 const int k = s.size();
3365 for (int i=0; i<=L2; i++) {
3366 crB.push_back(bernoulli(i).to_cl_N() / cln::factorial(i));
3371 for (int m=0; m<k-1; m++) {
3372 std::vector<cln::cl_N> crGbuf;
3375 for (int i=0; i<=L2; i++) {
3376 crGbuf.push_back(cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2));
3378 crG.push_back(crGbuf);
3383 for (int m=0; m<k-1; m++) {
3384 std::vector<cln::cl_N> Xbuf;
3385 for (int i=0; i<=L2; i++) {
3386 Xbuf.push_back(crX[i] * crG[m][i]);
3388 halfcyclic_convolute(Xbuf, crB, crX);
3394 cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk)
3396 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3397 cln::cl_N factor = cln::expt(lambda, Sqk);
3398 cln::cl_N res = factor / Sqk * crX[0] * one;
3403 factor = factor * lambda;
3405 res = res + crX[N] * factor / (N+Sqk);
3406 } while ((res != resbuf) || cln::zerop(crX[N]));
3412 void calc_f(int maxr)
3417 cln::cl_N t0, t1, t2, t3, t4;
3419 std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
3420 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3422 t0 = cln::exp(-lambda);
3424 for (k=1; k<=L1; k++) {
3427 for (j=1; j<=maxr; j++) {
3430 for (i=2; i<=j; i++) {
3434 (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
3442 cln::cl_N crandall_Z(const std::vector<int>& s)
3444 const int j = s.size();
3453 t0 = t0 + f_kj[q+j-2][s[0]-1];
3454 } while (t0 != t0buf);
3456 return t0 / cln::factorial(s[0]-1);
3459 std::vector<cln::cl_N> t(j);
3466 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
3467 for (int k=j-2; k>=1; k--) {
3468 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
3470 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
3471 } while (t[0] != t0buf);
3473 return t[0] / cln::factorial(s[0]-1);
3478 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
3480 std::vector<int> r = s;
3481 const int j = r.size();
3483 // decide on maximal size of f_kj for crandall_Z
3487 L1 = Digits * 3 + j*2;
3490 // decide on maximal size of crX for crandall_Y
3493 } else if (Digits < 86) {
3495 } else if (Digits < 192) {
3497 } else if (Digits < 394) {
3499 } else if (Digits < 808) {
3509 for (int i=0; i<j; i++) {
3518 const cln::cl_N r0factorial = cln::factorial(r[0]-1);
3520 std::vector<int> rz;
3523 for (int k=r.size()-1; k>0; k--) {
3525 rz.insert(rz.begin(), r.back());
3526 skp1buf = rz.front();
3532 for (int q=0; q<skp1buf; q++) {
3534 cln::cl_N pp1 = crandall_Y_loop(Srun+q-k);
3535 cln::cl_N pp2 = crandall_Z(rz);
3540 res = res - pp1 * pp2 / cln::factorial(q);
3542 res = res + pp1 * pp2 / cln::factorial(q);
3545 rz.front() = skp1buf;
3547 rz.insert(rz.begin(), r.back());
3551 res = (res + crandall_Y_loop(S-j)) / r0factorial + crandall_Z(rz);
3557 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
3559 const int j = r.size();
3561 // buffer for subsums
3562 std::vector<cln::cl_N> t(j);
3563 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3570 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
3571 for (int k=j-2; k>=0; k--) {
3572 t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
3574 } while (t[0] != t0buf);
3580 // does Hoelder convolution. see [BBB] (7.0)
3581 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
3583 // prepare parameters
3584 // holds Li arguments in [BBB] notation
3585 std::vector<int> s = s_;
3586 std::vector<int> m_p = m_;
3587 std::vector<int> m_q;
3588 // holds Li arguments in nested sums notation
3589 std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
3590 s_p[0] = s_p[0] * cln::cl_N("1/2");
3591 // convert notations
3593 for (int i=0; i<s_.size(); i++) {
3598 s[i] = sig * std::abs(s[i]);
3600 std::vector<cln::cl_N> s_q;
3601 cln::cl_N signum = 1;
3604 cln::cl_N res = multipleLi_do_sum(m_p, s_p);
3609 // change parameters
3610 if (s.front() > 0) {
3611 if (m_p.front() == 1) {
3612 m_p.erase(m_p.begin());
3613 s_p.erase(s_p.begin());
3614 if (s_p.size() > 0) {
3615 s_p.front() = s_p.front() * cln::cl_N("1/2");
3621 m_q.insert(m_q.begin(), 1);
3622 if (s_q.size() > 0) {
3623 s_q.front() = s_q.front() * 2;
3625 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3628 if (m_p.front() == 1) {
3629 m_p.erase(m_p.begin());
3630 cln::cl_N spbuf = s_p.front();
3631 s_p.erase(s_p.begin());
3632 if (s_p.size() > 0) {
3633 s_p.front() = s_p.front() * spbuf;
3636 m_q.insert(m_q.begin(), 1);
3637 if (s_q.size() > 0) {
3638 s_q.front() = s_q.front() * 4;
3640 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
3644 m_q.insert(m_q.begin(), 1);
3645 if (s_q.size() > 0) {
3646 s_q.front() = s_q.front() * 2;
3648 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3653 if (m_p.size() == 0) break;
3655 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
3660 res = res + signum * multipleLi_do_sum(m_q, s_q);
3666 } // end of anonymous namespace
3669 //////////////////////////////////////////////////////////////////////
3671 // Multiple zeta values zeta(x)
3675 //////////////////////////////////////////////////////////////////////
3678 static ex zeta1_evalf(const ex& x)
3680 if (is_exactly_a<lst>(x) && (x.nops()>1)) {
3682 // multiple zeta value
3683 const int count = x.nops();
3684 const lst& xlst = ex_to<lst>(x);
3685 std::vector<int> r(count);
3687 // check parameters and convert them
3688 lst::const_iterator it1 = xlst.begin();
3689 std::vector<int>::iterator it2 = r.begin();
3691 if (!(*it1).info(info_flags::posint)) {
3692 return zeta(x).hold();
3694 *it2 = ex_to<numeric>(*it1).to_int();
3697 } while (it2 != r.end());
3699 // check for divergence
3701 return zeta(x).hold();
3704 // decide on summation algorithm
3705 // this is still a bit clumsy
3706 int limit = (Digits>17) ? 10 : 6;
3707 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
3708 return numeric(zeta_do_sum_Crandall(r));
3710 return numeric(zeta_do_sum_simple(r));
3714 // single zeta value
3715 if (is_exactly_a<numeric>(x) && (x != 1)) {
3717 return zeta(ex_to<numeric>(x));
3718 } catch (const dunno &e) { }
3721 return zeta(x).hold();
3725 static ex zeta1_eval(const ex& m)
3727 if (is_exactly_a<lst>(m)) {
3728 if (m.nops() == 1) {
3729 return zeta(m.op(0));
3731 return zeta(m).hold();
3734 if (m.info(info_flags::numeric)) {
3735 const numeric& y = ex_to<numeric>(m);
3736 // trap integer arguments:
3737 if (y.is_integer()) {
3741 if (y.is_equal(*_num1_p)) {
3742 return zeta(m).hold();
3744 if (y.info(info_flags::posint)) {
3745 if (y.info(info_flags::odd)) {
3746 return zeta(m).hold();
3748 return abs(bernoulli(y)) * pow(Pi, y) * pow(*_num2_p, y-(*_num1_p)) / factorial(y);
3751 if (y.info(info_flags::odd)) {
3752 return -bernoulli((*_num1_p)-y) / ((*_num1_p)-y);
3759 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
3760 return zeta1_evalf(m);
3763 return zeta(m).hold();
3767 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
3769 GINAC_ASSERT(deriv_param==0);
3771 if (is_exactly_a<lst>(m)) {
3774 return zetaderiv(_ex1, m);
3779 static void zeta1_print_latex(const ex& m_, const print_context& c)
3782 if (is_a<lst>(m_)) {
3783 const lst& m = ex_to<lst>(m_);
3784 lst::const_iterator it = m.begin();
3787 for (; it != m.end(); it++) {
3798 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1).
3799 evalf_func(zeta1_evalf).
3800 eval_func(zeta1_eval).
3801 derivative_func(zeta1_deriv).
3802 print_func<print_latex>(zeta1_print_latex).
3803 do_not_evalf_params().
3807 //////////////////////////////////////////////////////////////////////
3809 // Alternating Euler sum zeta(x,s)
3813 //////////////////////////////////////////////////////////////////////
3816 static ex zeta2_evalf(const ex& x, const ex& s)
3818 if (is_exactly_a<lst>(x)) {
3820 // alternating Euler sum
3821 const int count = x.nops();
3822 const lst& xlst = ex_to<lst>(x);
3823 const lst& slst = ex_to<lst>(s);
3824 std::vector<int> xi(count);
3825 std::vector<int> si(count);
3827 // check parameters and convert them
3828 lst::const_iterator it_xread = xlst.begin();
3829 lst::const_iterator it_sread = slst.begin();
3830 std::vector<int>::iterator it_xwrite = xi.begin();
3831 std::vector<int>::iterator it_swrite = si.begin();
3833 if (!(*it_xread).info(info_flags::posint)) {
3834 return zeta(x, s).hold();
3836 *it_xwrite = ex_to<numeric>(*it_xread).to_int();
3837 if (*it_sread > 0) {
3846 } while (it_xwrite != xi.end());
3848 // check for divergence
3849 if ((xi[0] == 1) && (si[0] == 1)) {
3850 return zeta(x, s).hold();
3853 // use Hoelder convolution
3854 return numeric(zeta_do_Hoelder_convolution(xi, si));
3857 return zeta(x, s).hold();
3861 static ex zeta2_eval(const ex& m, const ex& s_)
3863 if (is_exactly_a<lst>(s_)) {
3864 const lst& s = ex_to<lst>(s_);
3865 for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
3866 if ((*it).info(info_flags::positive)) {
3869 return zeta(m, s_).hold();
3872 } else if (s_.info(info_flags::positive)) {
3876 return zeta(m, s_).hold();
3880 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
3882 GINAC_ASSERT(deriv_param==0);
3884 if (is_exactly_a<lst>(m)) {
3887 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
3888 return zetaderiv(_ex1, m);
3895 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
3898 if (is_a<lst>(m_)) {
3904 if (is_a<lst>(s_)) {
3910 lst::const_iterator itm = m.begin();
3911 lst::const_iterator its = s.begin();
3913 c.s << "\\overline{";
3921 for (; itm != m.end(); itm++, its++) {
3924 c.s << "\\overline{";
3935 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2).
3936 evalf_func(zeta2_evalf).
3937 eval_func(zeta2_eval).
3938 derivative_func(zeta2_deriv).
3939 print_func<print_latex>(zeta2_print_latex).
3940 do_not_evalf_params().
3944 } // namespace GiNaC