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) 1999-2008 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 const cln::cl_N S_num(int n, int p, const cln::cl_N& 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) - S_num(1, n-j-1, 1-x))
375 * cln::expt(cln::log(x), j) / cln::factorial(j);
382 // helper function for classical polylog Li
383 const cln::cl_N Lin_numeric(const int n, const cln::cl_N& x)
387 return -cln::log(1-x);
398 return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
400 if (cln::abs(realpart(x)) < 0.4 && cln::abs(cln::abs(x)-1) < 0.01) {
401 cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
402 for (int j=0; j<n-1; j++) {
403 result = result + (S_num(n-j-1, 1, 1) - S_num(1, n-j-1, 1-x))
404 * cln::expt(cln::log(x), j) / cln::factorial(j);
409 // what is the desired float format?
410 // first guess: default format
411 cln::float_format_t prec = cln::default_float_format;
412 const cln::cl_N value = x;
413 // second guess: the argument's format
414 if (!instanceof(realpart(x), cln::cl_RA_ring))
415 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
416 else if (!instanceof(imagpart(x), cln::cl_RA_ring))
417 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
420 if (cln::abs(value) > 1) {
421 cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
422 // check if argument is complex. if it is real, the new polylog has to be conjugated.
423 if (cln::zerop(cln::imagpart(value))) {
425 result = result + conjugate(Li_projection(n, cln::recip(value), prec));
428 result = result - conjugate(Li_projection(n, cln::recip(value), prec));
433 result = result + Li_projection(n, cln::recip(value), prec);
436 result = result - Li_projection(n, cln::recip(value), prec);
440 for (int j=0; j<n-1; j++) {
441 add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
442 * Lin_numeric(n-j,1) * cln::expt(cln::log(-value),j) / cln::factorial(j);
444 result = result - add;
448 return Li_projection(n, value, prec);
453 } // end of anonymous namespace
456 //////////////////////////////////////////////////////////////////////
458 // Multiple polylogarithm Li(n,x)
462 //////////////////////////////////////////////////////////////////////
465 // anonymous namespace for helper function
469 // performs the actual series summation for multiple polylogarithms
470 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
472 // ensure all x <> 0.
473 for (std::vector<cln::cl_N>::const_iterator it = x.begin(); it != x.end(); ++it) {
474 if ( *it == 0 ) return cln::cl_float(0, cln::float_format(Digits));
477 const int j = s.size();
478 bool flag_accidental_zero = false;
480 std::vector<cln::cl_N> t(j);
481 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
488 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
489 for (int k=j-2; k>=0; k--) {
490 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]);
493 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
494 for (int k=j-2; k>=0; k--) {
495 flag_accidental_zero = cln::zerop(t[k+1]);
496 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]);
498 } while ( (t[0] != t0buf) || cln::zerop(t[0]) || flag_accidental_zero );
504 // converts parameter types and calls multipleLi_do_sum (convenience function for G_numeric)
505 cln::cl_N mLi_do_summation(const lst& m, const lst& x)
507 std::vector<int> m_int;
508 std::vector<cln::cl_N> x_cln;
509 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
510 m_int.push_back(ex_to<numeric>(*itm).to_int());
511 x_cln.push_back(ex_to<numeric>(*itx).to_cl_N());
513 return multipleLi_do_sum(m_int, x_cln);
517 // forward declaration for Li_eval()
518 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
521 // holding dummy-symbols for the G/Li transformations
522 std::vector<ex> gsyms;
525 // type used by the transformation functions for G
526 typedef std::vector<int> Gparameter;
529 // G_eval1-function for G transformations
530 ex G_eval1(int a, int scale)
533 const ex& scs = gsyms[std::abs(scale)];
534 const ex& as = gsyms[std::abs(a)];
536 return -log(1 - scs/as);
541 return log(gsyms[std::abs(scale)]);
546 // G_eval-function for G transformations
547 ex G_eval(const Gparameter& a, int scale)
549 // check for properties of G
550 ex sc = gsyms[std::abs(scale)];
552 bool all_zero = true;
553 bool all_ones = true;
555 for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
557 const ex sym = gsyms[std::abs(*it)];
571 // care about divergent G: shuffle to separate divergencies that will be canceled
572 // later on in the transformation
573 if (newa.nops() > 1 && newa.op(0) == sc && !all_ones && a.front()!=0) {
576 Gparameter::const_iterator it = a.begin();
578 for (; it != a.end(); ++it) {
579 short_a.push_back(*it);
581 ex result = G_eval1(a.front(), scale) * G_eval(short_a, scale);
582 it = short_a.begin();
583 for (int i=1; i<count_ones; ++i) {
586 for (; it != short_a.end(); ++it) {
589 Gparameter::const_iterator it2 = short_a.begin();
590 for (--it2; it2 != it;) {
592 newa.push_back(*it2);
594 newa.push_back(a[0]);
596 for (; it2 != short_a.end(); ++it2) {
597 newa.push_back(*it2);
599 result -= G_eval(newa, scale);
601 return result / count_ones;
604 // G({1,...,1};y) -> G({1};y)^k / k!
605 if (all_ones && a.size() > 1) {
606 return pow(G_eval1(a.front(),scale), count_ones) / factorial(count_ones);
609 // G({0,...,0};y) -> log(y)^k / k!
611 return pow(log(gsyms[std::abs(scale)]), a.size()) / factorial(a.size());
614 // no special cases anymore -> convert it into Li
617 ex argbuf = gsyms[std::abs(scale)];
619 for (Gparameter::const_iterator it=a.begin(); it!=a.end(); ++it) {
621 const ex& sym = gsyms[std::abs(*it)];
622 x.append(argbuf / sym);
630 return pow(-1, x.nops()) * Li(m, x);
634 // converts data for G: pending_integrals -> a
635 Gparameter convert_pending_integrals_G(const Gparameter& pending_integrals)
637 GINAC_ASSERT(pending_integrals.size() != 1);
639 if (pending_integrals.size() > 0) {
640 // get rid of the first element, which would stand for the new upper limit
641 Gparameter new_a(pending_integrals.begin()+1, pending_integrals.end());
644 // just return empty parameter list
651 // check the parameters a and scale for G and return information about convergence, depth, etc.
652 // convergent : true if G(a,scale) is convergent
653 // depth : depth of G(a,scale)
654 // trailing_zeros : number of trailing zeros of a
655 // min_it : iterator of a pointing on the smallest element in a
656 Gparameter::const_iterator check_parameter_G(const Gparameter& a, int scale,
657 bool& convergent, int& depth, int& trailing_zeros, Gparameter::const_iterator& min_it)
663 Gparameter::const_iterator lastnonzero = a.end();
664 for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
665 if (std::abs(*it) > 0) {
669 if (std::abs(*it) < scale) {
671 if ((min_it == a.end()) || (std::abs(*it) < std::abs(*min_it))) {
679 return ++lastnonzero;
683 // add scale to pending_integrals if pending_integrals is empty
684 Gparameter prepare_pending_integrals(const Gparameter& pending_integrals, int scale)
686 GINAC_ASSERT(pending_integrals.size() != 1);
688 if (pending_integrals.size() > 0) {
689 return pending_integrals;
691 Gparameter new_pending_integrals;
692 new_pending_integrals.push_back(scale);
693 return new_pending_integrals;
698 // handles trailing zeroes for an otherwise convergent integral
699 ex trailing_zeros_G(const Gparameter& a, int scale)
702 int depth, trailing_zeros;
703 Gparameter::const_iterator last, dummyit;
704 last = check_parameter_G(a, scale, convergent, depth, trailing_zeros, dummyit);
706 GINAC_ASSERT(convergent);
708 if ((trailing_zeros > 0) && (depth > 0)) {
710 Gparameter new_a(a.begin(), a.end()-1);
711 result += G_eval1(0, scale) * trailing_zeros_G(new_a, scale);
712 for (Gparameter::const_iterator it = a.begin(); it != last; ++it) {
713 Gparameter new_a(a.begin(), it);
715 new_a.insert(new_a.end(), it, a.end()-1);
716 result -= trailing_zeros_G(new_a, scale);
719 return result / trailing_zeros;
721 return G_eval(a, scale);
726 // G transformation [VSW] (57),(58)
727 ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, int scale)
729 // pendint = ( y1, b1, ..., br )
730 // a = ( 0, ..., 0, amin )
733 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(0, ..., 0, sr; y2)
734 // where sr replaces amin
736 GINAC_ASSERT(a.back() != 0);
737 GINAC_ASSERT(a.size() > 0);
740 Gparameter new_pending_integrals = prepare_pending_integrals(pending_integrals, std::abs(a.back()));
741 const int psize = pending_integrals.size();
744 // G(sr_{+-}; y2 ) = G(y2_{-+}; sr) - G(0; sr) + ln(-y2_{-+})
749 result += log(gsyms[ex_to<numeric>(scale).to_int()]);
751 new_pending_integrals.push_back(-scale);
754 new_pending_integrals.push_back(scale);
758 result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals), pending_integrals.front());
762 result += trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals), new_pending_integrals.front());
765 new_pending_integrals.back() = 0;
766 result -= trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals), new_pending_integrals.front());
772 // G_m(sr_{+-}; y2) = -zeta_m + int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
773 // - int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
776 result -= zeta(a.size());
778 result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals), pending_integrals.front());
781 // term int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
782 // = int_0^sr dt/t G_{m-1}( t_{+-}; y2 )
783 Gparameter new_a(a.begin()+1, a.end());
784 new_pending_integrals.push_back(0);
785 result -= depth_one_trafo_G(new_pending_integrals, new_a, scale);
787 // term int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
788 // = int_0^y2 dt/t G_{m-1}( t_{+-}; y2 )
789 Gparameter new_pending_integrals_2;
790 new_pending_integrals_2.push_back(scale);
791 new_pending_integrals_2.push_back(0);
793 result += trailing_zeros_G(convert_pending_integrals_G(pending_integrals), pending_integrals.front())
794 * depth_one_trafo_G(new_pending_integrals_2, new_a, scale);
796 result += depth_one_trafo_G(new_pending_integrals_2, new_a, scale);
803 // forward declaration
804 ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
805 const Gparameter& pendint, const Gparameter& a_old, int scale);
808 // G transformation [VSW]
809 ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale)
811 // main recursion routine
813 // pendint = ( y1, b1, ..., br )
814 // a = ( a1, ..., amin, ..., aw )
817 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
818 // where sr replaces amin
820 // find smallest alpha, determine depth and trailing zeros, and check for convergence
822 int depth, trailing_zeros;
823 Gparameter::const_iterator min_it;
824 Gparameter::const_iterator firstzero =
825 check_parameter_G(a, scale, convergent, depth, trailing_zeros, min_it);
826 int min_it_pos = min_it - a.begin();
828 // special case: all a's are zero
835 result = G_eval(a, scale);
837 if (pendint.size() > 0) {
838 result *= trailing_zeros_G(convert_pending_integrals_G(pendint), pendint.front());
843 // handle trailing zeros
844 if (trailing_zeros > 0) {
846 Gparameter new_a(a.begin(), a.end()-1);
847 result += G_eval1(0, scale) * G_transform(pendint, new_a, scale);
848 for (Gparameter::const_iterator it = a.begin(); it != firstzero; ++it) {
849 Gparameter new_a(a.begin(), it);
851 new_a.insert(new_a.end(), it, a.end()-1);
852 result -= G_transform(pendint, new_a, scale);
854 return result / trailing_zeros;
859 if (pendint.size() > 0) {
860 return G_eval(convert_pending_integrals_G(pendint), pendint.front()) * G_eval(a, scale);
862 return G_eval(a, scale);
866 // call basic transformation for depth equal one
868 return depth_one_trafo_G(pendint, a, scale);
872 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
873 // = int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,0,...,aw,y2)
874 // + 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)
876 // smallest element in last place
877 if (min_it + 1 == a.end()) {
878 do { --min_it; } while (*min_it == 0);
880 Gparameter a1(a.begin(),min_it+1);
881 Gparameter a2(min_it+1,a.end());
883 ex result = G_transform(pendint,a2,scale)*G_transform(empty,a1,scale);
885 result -= shuffle_G(empty,a1,a2,pendint,a,scale);
890 Gparameter::iterator changeit;
892 // first term G(a_1,..,0,...,a_w;a_0)
893 Gparameter new_pendint = prepare_pending_integrals(pendint, a[min_it_pos]);
894 Gparameter new_a = a;
895 new_a[min_it_pos] = 0;
896 ex result = G_transform(empty, new_a, scale);
897 if (pendint.size() > 0) {
898 result *= trailing_zeros_G(convert_pending_integrals_G(pendint), pendint.front());
902 changeit = new_a.begin() + min_it_pos;
903 changeit = new_a.erase(changeit);
904 if (changeit != new_a.begin()) {
905 // smallest in the middle
906 new_pendint.push_back(*changeit);
907 result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
908 * G_transform(empty, new_a, scale);
909 int buffer = *changeit;
911 result += G_transform(new_pendint, new_a, scale);
913 new_pendint.pop_back();
915 new_pendint.push_back(*changeit);
916 result += trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
917 * G_transform(empty, new_a, scale);
919 result -= G_transform(new_pendint, new_a, scale);
921 // smallest at the front
922 new_pendint.push_back(scale);
923 result += trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
924 * G_transform(empty, new_a, scale);
925 new_pendint.back() = *changeit;
926 result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
927 * G_transform(empty, new_a, scale);
929 result += G_transform(new_pendint, new_a, scale);
935 // shuffles the two parameter list a1 and a2 and calls G_transform for every term except
936 // for the one that is equal to a_old
937 ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
938 const Gparameter& pendint, const Gparameter& a_old, int scale)
940 if (a1.size()==0 && a2.size()==0) {
941 // veto the one configuration we don't want
942 if ( a0 == a_old ) return 0;
944 return G_transform(pendint,a0,scale);
950 aa0.insert(aa0.end(),a1.begin(),a1.end());
951 return shuffle_G(aa0,empty,empty,pendint,a_old,scale);
957 aa0.insert(aa0.end(),a2.begin(),a2.end());
958 return shuffle_G(aa0,empty,empty,pendint,a_old,scale);
961 Gparameter a1_removed(a1.begin()+1,a1.end());
962 Gparameter a2_removed(a2.begin()+1,a2.end());
967 a01.push_back( a1[0] );
968 a02.push_back( a2[0] );
970 return shuffle_G(a01,a1_removed,a2,pendint,a_old,scale)
971 + shuffle_G(a02,a1,a2_removed,pendint,a_old,scale);
975 // handles the transformations and the numerical evaluation of G
976 // the parameter x, s and y must only contain numerics
977 ex G_numeric(const lst& x, const lst& s, const ex& y)
979 // check for convergence and necessary accelerations
980 bool need_trafo = false;
981 bool need_hoelder = false;
983 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
984 if (!(*it).is_zero()) {
986 if (abs(*it) - y < -pow(10,-Digits+1)) {
989 if (abs((abs(*it) - y)/y) < 0.01) {
994 if (x.op(x.nops()-1).is_zero()) {
997 if (depth == 1 && x.nops() == 2 && !need_trafo) {
998 return -Li(x.nops(), y / x.op(x.nops()-1)).evalf();
1001 // do acceleration transformation (hoelder convolution [BBB])
1005 const int size = x.nops();
1007 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1008 newx.append(*it / y);
1011 for (int r=0; r<=size; ++r) {
1012 ex buffer = pow(-1, r);
1017 for (lst::const_iterator it = newx.begin(); it != newx.end(); ++it) {
1028 for (int j=r; j>=1; --j) {
1029 qlstx.append(1-newx.op(j-1));
1030 if (newx.op(j-1).info(info_flags::real) && newx.op(j-1) > 1 && newx.op(j-1) <= 2) {
1031 qlsts.append( s.op(j-1));
1033 qlsts.append( -s.op(j-1));
1036 if (qlstx.nops() > 0) {
1037 buffer *= G_numeric(qlstx, qlsts, 1/q);
1041 for (int j=r+1; j<=size; ++j) {
1042 plstx.append(newx.op(j-1));
1043 plsts.append(s.op(j-1));
1045 if (plstx.nops() > 0) {
1046 buffer *= G_numeric(plstx, plsts, 1/p);
1053 // convergence transformation
1056 // sort (|x|<->position) to determine indices
1057 std::multimap<ex,int> sortmap;
1059 for (int i=0; i<x.nops(); ++i) {
1060 if (!x[i].is_zero()) {
1061 sortmap.insert(std::pair<ex,int>(abs(x[i]), i));
1065 // include upper limit (scale)
1066 sortmap.insert(std::pair<ex,int>(abs(y), x.nops()));
1068 // generate missing dummy-symbols
1071 gsyms.push_back(symbol("GSYMS_ERROR"));
1073 for (std::multimap<ex,int>::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1074 if (it != sortmap.begin()) {
1075 if (it->second < x.nops()) {
1076 if (x[it->second] == lastentry) {
1077 gsyms.push_back(gsyms.back());
1081 if (y == lastentry) {
1082 gsyms.push_back(gsyms.back());
1087 std::ostringstream os;
1089 gsyms.push_back(symbol(os.str()));
1091 if (it->second < x.nops()) {
1092 lastentry = x[it->second];
1098 // fill position data according to sorted indices and prepare substitution list
1099 Gparameter a(x.nops());
1103 for (std::multimap<ex,int>::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1104 if (it->second < x.nops()) {
1105 if (s[it->second] > 0) {
1106 a[it->second] = pos;
1108 a[it->second] = -pos;
1110 subslst.append(gsyms[pos] == x[it->second]);
1113 subslst.append(gsyms[pos] == y);
1118 // do transformation
1120 ex result = G_transform(pendint, a, scale);
1121 // replace dummy symbols with their values
1122 result = result.eval().expand();
1123 result = result.subs(subslst).evalf();
1134 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1135 if ((*it).is_zero()) {
1138 newx.append(factor / (*it));
1146 return sign * numeric(mLi_do_summation(m, newx));
1150 ex mLi_numeric(const lst& m, const lst& x)
1152 // let G_numeric do the transformation
1156 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1157 for (int i = 1; i < *itm; ++i) {
1161 newx.append(factor / *itx);
1165 return pow(-1, m.nops()) * G_numeric(newx, s, _ex1);
1169 } // end of anonymous namespace
1172 //////////////////////////////////////////////////////////////////////
1174 // Generalized multiple polylogarithm G(x, y) and G(x, s, y)
1178 //////////////////////////////////////////////////////////////////////
1181 static ex G2_evalf(const ex& x_, const ex& y)
1183 if (!y.info(info_flags::positive)) {
1184 return G(x_, y).hold();
1186 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1187 if (x.nops() == 0) {
1191 return G(x_, y).hold();
1194 bool all_zero = true;
1195 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1196 if (!(*it).info(info_flags::numeric)) {
1197 return G(x_, y).hold();
1205 return pow(log(y), x.nops()) / factorial(x.nops());
1207 return G_numeric(x, s, y);
1211 static ex G2_eval(const ex& x_, const ex& y)
1213 //TODO eval to MZV or H or S or Lin
1215 if (!y.info(info_flags::positive)) {
1216 return G(x_, y).hold();
1218 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1219 if (x.nops() == 0) {
1223 return G(x_, y).hold();
1226 bool all_zero = true;
1227 bool crational = true;
1228 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1229 if (!(*it).info(info_flags::numeric)) {
1230 return G(x_, y).hold();
1232 if (!(*it).info(info_flags::crational)) {
1241 return pow(log(y), x.nops()) / factorial(x.nops());
1243 if (!y.info(info_flags::crational)) {
1247 return G(x_, y).hold();
1249 return G_numeric(x, s, y);
1253 unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2).
1254 evalf_func(G2_evalf).
1256 do_not_evalf_params().
1259 // derivative_func(G2_deriv).
1260 // print_func<print_latex>(G2_print_latex).
1263 static ex G3_evalf(const ex& x_, const ex& s_, const ex& y)
1265 if (!y.info(info_flags::positive)) {
1266 return G(x_, s_, y).hold();
1268 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1269 lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
1270 if (x.nops() != s.nops()) {
1271 return G(x_, s_, y).hold();
1273 if (x.nops() == 0) {
1277 return G(x_, s_, y).hold();
1280 bool all_zero = true;
1281 for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1282 if (!(*itx).info(info_flags::numeric)) {
1283 return G(x_, y).hold();
1285 if (!(*its).info(info_flags::real)) {
1286 return G(x_, y).hold();
1298 return pow(log(y), x.nops()) / factorial(x.nops());
1300 return G_numeric(x, sn, y);
1304 static ex G3_eval(const ex& x_, const ex& s_, const ex& y)
1306 //TODO eval to MZV or H or S or Lin
1308 if (!y.info(info_flags::positive)) {
1309 return G(x_, s_, y).hold();
1311 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1312 lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
1313 if (x.nops() != s.nops()) {
1314 return G(x_, s_, y).hold();
1316 if (x.nops() == 0) {
1320 return G(x_, s_, y).hold();
1323 bool all_zero = true;
1324 bool crational = true;
1325 for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1326 if (!(*itx).info(info_flags::numeric)) {
1327 return G(x_, s_, y).hold();
1329 if (!(*its).info(info_flags::real)) {
1330 return G(x_, s_, y).hold();
1332 if (!(*itx).info(info_flags::crational)) {
1345 return pow(log(y), x.nops()) / factorial(x.nops());
1347 if (!y.info(info_flags::crational)) {
1351 return G(x_, s_, y).hold();
1353 return G_numeric(x, sn, y);
1357 unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3).
1358 evalf_func(G3_evalf).
1360 do_not_evalf_params().
1363 // derivative_func(G3_deriv).
1364 // print_func<print_latex>(G3_print_latex).
1367 //////////////////////////////////////////////////////////////////////
1369 // Classical polylogarithm and multiple polylogarithm Li(m,x)
1373 //////////////////////////////////////////////////////////////////////
1376 static ex Li_evalf(const ex& m_, const ex& x_)
1378 // classical polylogs
1379 if (m_.info(info_flags::posint)) {
1380 if (x_.info(info_flags::numeric)) {
1381 int m__ = ex_to<numeric>(m_).to_int();
1382 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1383 const cln::cl_N result = Lin_numeric(m__, x__);
1384 return numeric(result);
1386 // try to numerically evaluate second argument
1387 ex x_val = x_.evalf();
1388 if (x_val.info(info_flags::numeric)) {
1389 int m__ = ex_to<numeric>(m_).to_int();
1390 const cln::cl_N x__ = ex_to<numeric>(x_val).to_cl_N();
1391 const cln::cl_N result = Lin_numeric(m__, x__);
1392 return numeric(result);
1396 // multiple polylogs
1397 if (is_a<lst>(m_) && is_a<lst>(x_)) {
1399 const lst& m = ex_to<lst>(m_);
1400 const lst& x = ex_to<lst>(x_);
1401 if (m.nops() != x.nops()) {
1402 return Li(m_,x_).hold();
1404 if (x.nops() == 0) {
1407 if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) {
1408 return Li(m_,x_).hold();
1411 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1412 if (!(*itm).info(info_flags::posint)) {
1413 return Li(m_, x_).hold();
1415 if (!(*itx).info(info_flags::numeric)) {
1416 return Li(m_, x_).hold();
1423 return mLi_numeric(m, x);
1426 return Li(m_,x_).hold();
1430 static ex Li_eval(const ex& m_, const ex& x_)
1432 if (is_a<lst>(m_)) {
1433 if (is_a<lst>(x_)) {
1434 // multiple polylogs
1435 const lst& m = ex_to<lst>(m_);
1436 const lst& x = ex_to<lst>(x_);
1437 if (m.nops() != x.nops()) {
1438 return Li(m_,x_).hold();
1440 if (x.nops() == 0) {
1444 bool is_zeta = true;
1445 bool do_evalf = true;
1446 bool crational = true;
1447 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1448 if (!(*itm).info(info_flags::posint)) {
1449 return Li(m_,x_).hold();
1451 if ((*itx != _ex1) && (*itx != _ex_1)) {
1452 if (itx != x.begin()) {
1460 if (!(*itx).info(info_flags::numeric)) {
1463 if (!(*itx).info(info_flags::crational)) {
1472 lst newm = convert_parameter_Li_to_H(m, x, prefactor);
1473 return prefactor * H(newm, x[0]);
1475 if (do_evalf && !crational) {
1476 return mLi_numeric(m,x);
1479 return Li(m_, x_).hold();
1480 } else if (is_a<lst>(x_)) {
1481 return Li(m_, x_).hold();
1484 // classical polylogs
1492 return (pow(2,1-m_)-1) * zeta(m_);
1498 if (x_.is_equal(I)) {
1499 return power(Pi,_ex2)/_ex_48 + Catalan*I;
1501 if (x_.is_equal(-I)) {
1502 return power(Pi,_ex2)/_ex_48 - Catalan*I;
1505 if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
1506 int m__ = ex_to<numeric>(m_).to_int();
1507 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1508 const cln::cl_N result = Lin_numeric(m__, x__);
1509 return numeric(result);
1512 return Li(m_, x_).hold();
1516 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
1518 if (is_a<lst>(m) || is_a<lst>(x)) {
1521 seq.push_back(expair(Li(m, x), 0));
1522 return pseries(rel, seq);
1525 // classical polylog
1526 const ex x_pt = x.subs(rel, subs_options::no_pattern);
1527 if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
1528 // First special case: x==0 (derivatives have poles)
1529 if (x_pt.is_zero()) {
1532 // manually construct the primitive expansion
1533 for (int i=1; i<order; ++i)
1534 ser += pow(s,i) / pow(numeric(i), m);
1535 // substitute the argument's series expansion
1536 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
1537 // maybe that was terminating, so add a proper order term
1539 nseq.push_back(expair(Order(_ex1), order));
1540 ser += pseries(rel, nseq);
1541 // reexpanding it will collapse the series again
1542 return ser.series(rel, order);
1544 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
1545 throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
1547 // all other cases should be safe, by now:
1548 throw do_taylor(); // caught by function::series()
1552 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
1554 GINAC_ASSERT(deriv_param < 2);
1555 if (deriv_param == 0) {
1558 if (m_.nops() > 1) {
1559 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
1562 if (is_a<lst>(m_)) {
1568 if (is_a<lst>(x_)) {
1574 return Li(m-1, x) / x;
1581 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
1584 if (is_a<lst>(m_)) {
1590 if (is_a<lst>(x_)) {
1595 c.s << "\\mbox{Li}_{";
1596 lst::const_iterator itm = m.begin();
1599 for (; itm != m.end(); itm++) {
1604 lst::const_iterator itx = x.begin();
1607 for (; itx != x.end(); itx++) {
1615 REGISTER_FUNCTION(Li,
1616 evalf_func(Li_evalf).
1618 series_func(Li_series).
1619 derivative_func(Li_deriv).
1620 print_func<print_latex>(Li_print_latex).
1621 do_not_evalf_params());
1624 //////////////////////////////////////////////////////////////////////
1626 // Nielsen's generalized polylogarithm S(n,p,x)
1630 //////////////////////////////////////////////////////////////////////
1633 // anonymous namespace for helper functions
1637 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
1639 std::vector<std::vector<cln::cl_N> > Yn;
1640 int ynsize = 0; // number of Yn[]
1641 int ynlength = 100; // initial length of all Yn[i]
1644 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
1645 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
1646 // representing S_{n,p}(x).
1647 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
1648 // equivalent Z-sum.
1649 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
1650 // representing S_{n,p}(x).
1651 // The calculation of Y_n uses the values from Y_{n-1}.
1652 void fill_Yn(int n, const cln::float_format_t& prec)
1654 const int initsize = ynlength;
1655 //const int initsize = initsize_Yn;
1656 cln::cl_N one = cln::cl_float(1, prec);
1659 std::vector<cln::cl_N> buf(initsize);
1660 std::vector<cln::cl_N>::iterator it = buf.begin();
1661 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1662 *it = (*itprev) / cln::cl_N(n+1) * one;
1665 // sums with an index smaller than the depth are zero and need not to be calculated.
1666 // calculation starts with depth, which is n+2)
1667 for (int i=n+2; i<=initsize+n; i++) {
1668 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1674 std::vector<cln::cl_N> buf(initsize);
1675 std::vector<cln::cl_N>::iterator it = buf.begin();
1678 for (int i=2; i<=initsize; i++) {
1679 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1688 // make Yn longer ...
1689 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
1692 cln::cl_N one = cln::cl_float(1, prec);
1694 Yn[0].resize(newsize);
1695 std::vector<cln::cl_N>::iterator it = Yn[0].begin();
1697 for (int i=ynlength+1; i<=newsize; i++) {
1698 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1702 for (int n=1; n<ynsize; n++) {
1703 Yn[n].resize(newsize);
1704 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
1705 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1708 for (int i=ynlength+n+1; i<=newsize+n; i++) {
1709 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1719 // helper function for S(n,p,x)
1721 cln::cl_N C(int n, int p)
1725 for (int k=0; k<p; k++) {
1726 for (int j=0; j<=(n+k-1)/2; j++) {
1730 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1733 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1740 result = result + cln::factorial(n+k-1)
1741 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1742 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1745 result = result - cln::factorial(n+k-1)
1746 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1747 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1752 result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1753 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1756 result = result + cln::factorial(n+k-1)
1757 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1758 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1766 if (((np)/2+n) & 1) {
1767 result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1770 result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1778 // helper function for S(n,p,x)
1779 // [Kol] remark to (9.1)
1780 cln::cl_N a_k(int k)
1789 for (int m=2; m<=k; m++) {
1790 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
1797 // helper function for S(n,p,x)
1798 // [Kol] remark to (9.1)
1799 cln::cl_N b_k(int k)
1808 for (int m=2; m<=k; m++) {
1809 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
1816 // helper function for S(n,p,x)
1817 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1819 static cln::float_format_t oldprec = cln::default_float_format;
1822 return Li_projection(n+1, x, prec);
1825 // precision has changed, we need to clear lookup table Yn
1826 if ( oldprec != prec ) {
1833 // check if precalculated values are sufficient
1835 for (int i=ynsize; i<p-1; i++) {
1840 // should be done otherwise
1841 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1842 cln::cl_N xf = x * one;
1843 //cln::cl_N xf = x * cln::cl_float(1, prec);
1847 cln::cl_N factor = cln::expt(xf, p);
1851 if (i-p >= ynlength) {
1853 make_Yn_longer(ynlength*2, prec);
1855 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? ...
1856 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
1857 factor = factor * xf;
1859 } while (res != resbuf);
1865 // helper function for S(n,p,x)
1866 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1869 if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
1871 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
1872 * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
1874 for (int s=0; s<n; s++) {
1876 for (int r=0; r<p; r++) {
1877 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
1878 * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
1880 result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
1886 return S_do_sum(n, p, x, prec);
1890 // helper function for S(n,p,x)
1891 const cln::cl_N S_num(int n, int p, const cln::cl_N& x)
1895 // [Kol] (2.22) with (2.21)
1896 return cln::zeta(p+1);
1901 return cln::zeta(n+1);
1906 for (int nu=0; nu<n; nu++) {
1907 for (int rho=0; rho<=p; rho++) {
1908 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
1909 * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
1912 result = result * cln::expt(cln::cl_I(-1),n+p-1);
1919 return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
1921 // throw std::runtime_error("don't know how to evaluate this function!");
1924 // what is the desired float format?
1925 // first guess: default format
1926 cln::float_format_t prec = cln::default_float_format;
1927 const cln::cl_N value = x;
1928 // second guess: the argument's format
1929 if (!instanceof(realpart(value), cln::cl_RA_ring))
1930 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
1931 else if (!instanceof(imagpart(value), cln::cl_RA_ring))
1932 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
1935 if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95))) {
1937 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
1938 * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
1940 for (int s=0; s<n; s++) {
1942 for (int r=0; r<p; r++) {
1943 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
1944 * S_num(p-r,n-s,1-value) / cln::factorial(r);
1946 result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
1953 if (cln::abs(value) > 1) {
1957 for (int s=0; s<p; s++) {
1958 for (int r=0; r<=s; r++) {
1959 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
1960 / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
1961 * S_num(n+s-r,p-s,cln::recip(value));
1964 result = result * cln::expt(cln::cl_I(-1),n);
1967 for (int r=0; r<n; r++) {
1968 res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
1970 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
1972 result = result + cln::expt(cln::cl_I(-1),p) * res2;
1977 return S_projection(n, p, value, prec);
1982 } // end of anonymous namespace
1985 //////////////////////////////////////////////////////////////////////
1987 // Nielsen's generalized polylogarithm S(n,p,x)
1991 //////////////////////////////////////////////////////////////////////
1994 static ex S_evalf(const ex& n, const ex& p, const ex& x)
1996 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
1997 const int n_ = ex_to<numeric>(n).to_int();
1998 const int p_ = ex_to<numeric>(p).to_int();
1999 if (is_a<numeric>(x)) {
2000 const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2001 const cln::cl_N result = S_num(n_, p_, x_);
2002 return numeric(result);
2004 ex x_val = x.evalf();
2005 if (is_a<numeric>(x_val)) {
2006 const cln::cl_N x_val_ = ex_to<numeric>(x_val).to_cl_N();
2007 const cln::cl_N result = S_num(n_, p_, x_val_);
2008 return numeric(result);
2012 return S(n, p, x).hold();
2016 static ex S_eval(const ex& n, const ex& p, const ex& x)
2018 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2024 for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
2032 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2033 int n_ = ex_to<numeric>(n).to_int();
2034 int p_ = ex_to<numeric>(p).to_int();
2035 const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2036 const cln::cl_N result = S_num(n_, p_, x_);
2037 return numeric(result);
2042 return pow(-log(1-x), p) / factorial(p);
2044 return S(n, p, x).hold();
2048 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
2051 return Li(n+1, x).series(rel, order, options);
2054 const ex x_pt = x.subs(rel, subs_options::no_pattern);
2055 if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
2056 // First special case: x==0 (derivatives have poles)
2057 if (x_pt.is_zero()) {
2060 // manually construct the primitive expansion
2061 // subsum = Euler-Zagier-Sum is needed
2062 // dirty hack (slow ...) calculation of subsum:
2063 std::vector<ex> presubsum, subsum;
2064 subsum.push_back(0);
2065 for (int i=1; i<order-1; ++i) {
2066 subsum.push_back(subsum[i-1] + numeric(1, i));
2068 for (int depth=2; depth<p; ++depth) {
2070 for (int i=1; i<order-1; ++i) {
2071 subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
2075 for (int i=1; i<order; ++i) {
2076 ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
2078 // substitute the argument's series expansion
2079 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
2080 // maybe that was terminating, so add a proper order term
2082 nseq.push_back(expair(Order(_ex1), order));
2083 ser += pseries(rel, nseq);
2084 // reexpanding it will collapse the series again
2085 return ser.series(rel, order);
2087 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
2088 throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
2090 // all other cases should be safe, by now:
2091 throw do_taylor(); // caught by function::series()
2095 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
2097 GINAC_ASSERT(deriv_param < 3);
2098 if (deriv_param < 2) {
2102 return S(n-1, p, x) / x;
2104 return S(n, p-1, x) / (1-x);
2109 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
2111 c.s << "\\mbox{S}_{";
2121 REGISTER_FUNCTION(S,
2122 evalf_func(S_evalf).
2124 series_func(S_series).
2125 derivative_func(S_deriv).
2126 print_func<print_latex>(S_print_latex).
2127 do_not_evalf_params());
2130 //////////////////////////////////////////////////////////////////////
2132 // Harmonic polylogarithm H(m,x)
2136 //////////////////////////////////////////////////////////////////////
2139 // anonymous namespace for helper functions
2143 // regulates the pole (used by 1/x-transformation)
2144 symbol H_polesign("IMSIGN");
2147 // convert parameters from H to Li representation
2148 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
2149 // returns true if some parameters are negative
2150 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
2152 // expand parameter list
2154 for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
2156 for (ex count=*it-1; count > 0; count--) {
2160 } else if (*it < -1) {
2161 for (ex count=*it+1; count < 0; count++) {
2172 bool has_negative_parameters = false;
2174 for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
2180 m.append((*it+acc-1) * signum);
2182 m.append((*it-acc+1) * signum);
2188 has_negative_parameters = true;
2191 if (has_negative_parameters) {
2192 for (int i=0; i<m.nops(); i++) {
2194 m.let_op(i) = -m.op(i);
2202 return has_negative_parameters;
2206 // recursivly transforms H to corresponding multiple polylogarithms
2207 struct map_trafo_H_convert_to_Li : public map_function
2209 ex operator()(const ex& e)
2211 if (is_a<add>(e) || is_a<mul>(e)) {
2212 return e.map(*this);
2214 if (is_a<function>(e)) {
2215 std::string name = ex_to<function>(e).get_name();
2218 if (is_a<lst>(e.op(0))) {
2219 parameter = ex_to<lst>(e.op(0));
2221 parameter = lst(e.op(0));
2228 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2229 s.let_op(0) = s.op(0) * arg;
2230 return pf * Li(m, s).hold();
2232 for (int i=0; i<m.nops(); i++) {
2235 s.let_op(0) = s.op(0) * arg;
2236 return Li(m, s).hold();
2245 // recursivly transforms H to corresponding zetas
2246 struct map_trafo_H_convert_to_zeta : public map_function
2248 ex operator()(const ex& e)
2250 if (is_a<add>(e) || is_a<mul>(e)) {
2251 return e.map(*this);
2253 if (is_a<function>(e)) {
2254 std::string name = ex_to<function>(e).get_name();
2257 if (is_a<lst>(e.op(0))) {
2258 parameter = ex_to<lst>(e.op(0));
2260 parameter = lst(e.op(0));
2266 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2267 return pf * zeta(m, s);
2278 // remove trailing zeros from H-parameters
2279 struct map_trafo_H_reduce_trailing_zeros : public map_function
2281 ex operator()(const ex& e)
2283 if (is_a<add>(e) || is_a<mul>(e)) {
2284 return e.map(*this);
2286 if (is_a<function>(e)) {
2287 std::string name = ex_to<function>(e).get_name();
2290 if (is_a<lst>(e.op(0))) {
2291 parameter = ex_to<lst>(e.op(0));
2293 parameter = lst(e.op(0));
2296 if (parameter.op(parameter.nops()-1) == 0) {
2299 if (parameter.nops() == 1) {
2304 lst::const_iterator it = parameter.begin();
2305 while ((it != parameter.end()) && (*it == 0)) {
2308 if (it == parameter.end()) {
2309 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
2313 parameter.remove_last();
2314 int lastentry = parameter.nops();
2315 while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
2320 ex result = log(arg) * H(parameter,arg).hold();
2322 for (ex i=0; i<lastentry; i++) {
2323 if (parameter[i] > 0) {
2325 result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
2328 } else if (parameter[i] < 0) {
2330 result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
2338 if (lastentry < parameter.nops()) {
2339 result = result / (parameter.nops()-lastentry+1);
2340 return result.map(*this);
2352 // returns an expression with zeta functions corresponding to the parameter list for H
2353 ex convert_H_to_zeta(const lst& m)
2355 symbol xtemp("xtemp");
2356 map_trafo_H_reduce_trailing_zeros filter;
2357 map_trafo_H_convert_to_zeta filter2;
2358 return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
2362 // convert signs form Li to H representation
2363 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
2366 lst::const_iterator itm = m.begin();
2367 lst::const_iterator itx = ++x.begin();
2372 while (itx != x.end()) {
2373 signum *= (*itx > 0) ? 1 : -1;
2375 res.append((*itm) * signum);
2383 // multiplies an one-dimensional H with another H
2385 ex trafo_H_mult(const ex& h1, const ex& h2)
2390 ex h1nops = h1.op(0).nops();
2391 ex h2nops = h2.op(0).nops();
2393 hshort = h2.op(0).op(0);
2394 hlong = ex_to<lst>(h1.op(0));
2396 hshort = h1.op(0).op(0);
2398 hlong = ex_to<lst>(h2.op(0));
2400 hlong = h2.op(0).op(0);
2403 for (int i=0; i<=hlong.nops(); i++) {
2407 newparameter.append(hlong[j]);
2409 newparameter.append(hshort);
2410 for (; j<hlong.nops(); j++) {
2411 newparameter.append(hlong[j]);
2413 res += H(newparameter, h1.op(1)).hold();
2419 // applies trafo_H_mult recursively on expressions
2420 struct map_trafo_H_mult : public map_function
2422 ex operator()(const ex& e)
2425 return e.map(*this);
2433 for (int pos=0; pos<e.nops(); pos++) {
2434 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
2435 std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
2437 for (ex i=0; i<e.op(pos).op(1); i++) {
2438 Hlst.append(e.op(pos).op(0));
2442 } else if (is_a<function>(e.op(pos))) {
2443 std::string name = ex_to<function>(e.op(pos)).get_name();
2445 if (e.op(pos).op(0).nops() > 1) {
2448 Hlst.append(e.op(pos));
2453 result *= e.op(pos);
2456 if (Hlst.nops() > 0) {
2457 firstH = Hlst[Hlst.nops()-1];
2464 if (Hlst.nops() > 0) {
2465 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
2467 for (int i=1; i<Hlst.nops(); i++) {
2468 result *= Hlst.op(i);
2470 result = result.expand();
2471 map_trafo_H_mult recursion;
2472 return recursion(result);
2483 // do integration [ReV] (55)
2484 // put parameter 0 in front of existing parameters
2485 ex trafo_H_1tx_prepend_zero(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(0);
2507 ex addzeta = convert_H_to_zeta(newparameter);
2508 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2510 return e * (-H(lst(0),1/arg).hold());
2515 // do integration [ReV] (49)
2516 // put parameter 1 in front of existing parameters
2517 ex trafo_H_prepend_one(const ex& e, const ex& arg)
2521 if (is_a<function>(e)) {
2522 name = ex_to<function>(e).get_name();
2527 for (int i=0; i<e.nops(); i++) {
2528 if (is_a<function>(e.op(i))) {
2529 std::string name = ex_to<function>(e.op(i)).get_name();
2537 lst newparameter = ex_to<lst>(h.op(0));
2538 newparameter.prepend(1);
2539 return e.subs(h == H(newparameter, h.op(1)).hold());
2541 return e * H(lst(1),1-arg).hold();
2546 // do integration [ReV] (55)
2547 // put parameter -1 in front of existing parameters
2548 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
2552 if (is_a<function>(e)) {
2553 name = ex_to<function>(e).get_name();
2558 for (int i=0; i<e.nops(); i++) {
2559 if (is_a<function>(e.op(i))) {
2560 std::string name = ex_to<function>(e.op(i)).get_name();
2568 lst newparameter = ex_to<lst>(h.op(0));
2569 newparameter.prepend(-1);
2570 ex addzeta = convert_H_to_zeta(newparameter);
2571 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2573 ex addzeta = convert_H_to_zeta(lst(-1));
2574 return (e * (addzeta - H(lst(-1),1/arg).hold())).expand();
2579 // do integration [ReV] (55)
2580 // put parameter -1 in front of existing parameters
2581 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
2585 if (is_a<function>(e)) {
2586 name = ex_to<function>(e).get_name();
2591 for (int i=0; i<e.nops(); i++) {
2592 if (is_a<function>(e.op(i))) {
2593 std::string name = ex_to<function>(e.op(i)).get_name();
2601 lst newparameter = ex_to<lst>(h.op(0));
2602 newparameter.prepend(-1);
2603 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2605 return (e * H(lst(-1),(1-arg)/(1+arg)).hold()).expand();
2610 // do integration [ReV] (55)
2611 // put parameter 1 in front of existing parameters
2612 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
2616 if (is_a<function>(e)) {
2617 name = ex_to<function>(e).get_name();
2622 for (int i=0; i<e.nops(); i++) {
2623 if (is_a<function>(e.op(i))) {
2624 std::string name = ex_to<function>(e.op(i)).get_name();
2632 lst newparameter = ex_to<lst>(h.op(0));
2633 newparameter.prepend(1);
2634 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2636 return (e * H(lst(1),(1-arg)/(1+arg)).hold()).expand();
2641 // do x -> 1-x transformation
2642 struct map_trafo_H_1mx : public map_function
2644 ex operator()(const ex& e)
2646 if (is_a<add>(e) || is_a<mul>(e)) {
2647 return e.map(*this);
2650 if (is_a<function>(e)) {
2651 std::string name = ex_to<function>(e).get_name();
2654 lst parameter = ex_to<lst>(e.op(0));
2657 // special cases if all parameters are either 0, 1 or -1
2658 bool allthesame = true;
2659 if (parameter.op(0) == 0) {
2660 for (int i=1; i<parameter.nops(); i++) {
2661 if (parameter.op(i) != 0) {
2668 for (int i=parameter.nops(); i>0; i--) {
2669 newparameter.append(1);
2671 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2673 } else if (parameter.op(0) == -1) {
2674 throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
2676 for (int i=1; i<parameter.nops(); i++) {
2677 if (parameter.op(i) != 1) {
2684 for (int i=parameter.nops(); i>0; i--) {
2685 newparameter.append(0);
2687 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2691 lst newparameter = parameter;
2692 newparameter.remove_first();
2694 if (parameter.op(0) == 0) {
2697 ex res = convert_H_to_zeta(parameter);
2698 //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
2699 map_trafo_H_1mx recursion;
2700 ex buffer = recursion(H(newparameter, arg).hold());
2701 if (is_a<add>(buffer)) {
2702 for (int i=0; i<buffer.nops(); i++) {
2703 res -= trafo_H_prepend_one(buffer.op(i), arg);
2706 res -= trafo_H_prepend_one(buffer, arg);
2713 map_trafo_H_1mx recursion;
2714 map_trafo_H_mult unify;
2715 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
2717 while (parameter.op(firstzero) == 1) {
2720 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
2724 newparameter.append(parameter[j+1]);
2726 newparameter.append(1);
2727 for (; j<parameter.nops()-1; j++) {
2728 newparameter.append(parameter[j+1]);
2730 res -= H(newparameter, arg).hold();
2732 res = recursion(res).expand() / firstzero;
2742 // do x -> 1/x transformation
2743 struct map_trafo_H_1overx : public map_function
2745 ex operator()(const ex& e)
2747 if (is_a<add>(e) || is_a<mul>(e)) {
2748 return e.map(*this);
2751 if (is_a<function>(e)) {
2752 std::string name = ex_to<function>(e).get_name();
2755 lst parameter = ex_to<lst>(e.op(0));
2758 // special cases if all parameters are either 0, 1 or -1
2759 bool allthesame = true;
2760 if (parameter.op(0) == 0) {
2761 for (int i=1; i<parameter.nops(); i++) {
2762 if (parameter.op(i) != 0) {
2768 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
2770 } else if (parameter.op(0) == -1) {
2771 for (int i=1; i<parameter.nops(); i++) {
2772 if (parameter.op(i) != -1) {
2778 map_trafo_H_mult unify;
2779 return unify((pow(H(lst(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops())
2780 / factorial(parameter.nops())).expand());
2783 for (int i=1; i<parameter.nops(); i++) {
2784 if (parameter.op(i) != 1) {
2790 map_trafo_H_mult unify;
2791 return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() + H_polesign, parameter.nops())
2792 / factorial(parameter.nops())).expand());
2796 lst newparameter = parameter;
2797 newparameter.remove_first();
2799 if (parameter.op(0) == 0) {
2802 ex res = convert_H_to_zeta(parameter);
2803 map_trafo_H_1overx recursion;
2804 ex buffer = recursion(H(newparameter, arg).hold());
2805 if (is_a<add>(buffer)) {
2806 for (int i=0; i<buffer.nops(); i++) {
2807 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
2810 res += trafo_H_1tx_prepend_zero(buffer, arg);
2814 } else if (parameter.op(0) == -1) {
2816 // leading negative one
2817 ex res = convert_H_to_zeta(parameter);
2818 map_trafo_H_1overx recursion;
2819 ex buffer = recursion(H(newparameter, arg).hold());
2820 if (is_a<add>(buffer)) {
2821 for (int i=0; i<buffer.nops(); i++) {
2822 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
2825 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
2832 map_trafo_H_1overx recursion;
2833 map_trafo_H_mult unify;
2834 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
2836 while (parameter.op(firstzero) == 1) {
2839 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
2843 newparameter.append(parameter[j+1]);
2845 newparameter.append(1);
2846 for (; j<parameter.nops()-1; j++) {
2847 newparameter.append(parameter[j+1]);
2849 res -= H(newparameter, arg).hold();
2851 res = recursion(res).expand() / firstzero;
2863 // do x -> (1-x)/(1+x) transformation
2864 struct map_trafo_H_1mxt1px : public map_function
2866 ex operator()(const ex& e)
2868 if (is_a<add>(e) || is_a<mul>(e)) {
2869 return e.map(*this);
2872 if (is_a<function>(e)) {
2873 std::string name = ex_to<function>(e).get_name();
2876 lst parameter = ex_to<lst>(e.op(0));
2879 // special cases if all parameters are either 0, 1 or -1
2880 bool allthesame = true;
2881 if (parameter.op(0) == 0) {
2882 for (int i=1; i<parameter.nops(); i++) {
2883 if (parameter.op(i) != 0) {
2889 map_trafo_H_mult unify;
2890 return unify((pow(-H(lst(1),(1-arg)/(1+arg)).hold() - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
2891 / factorial(parameter.nops())).expand());
2893 } else if (parameter.op(0) == -1) {
2894 for (int i=1; i<parameter.nops(); i++) {
2895 if (parameter.op(i) != -1) {
2901 map_trafo_H_mult unify;
2902 return unify((pow(log(2) - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
2903 / factorial(parameter.nops())).expand());
2906 for (int i=1; i<parameter.nops(); i++) {
2907 if (parameter.op(i) != 1) {
2913 map_trafo_H_mult unify;
2914 return unify((pow(-log(2) - H(lst(0),(1-arg)/(1+arg)).hold() + H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
2915 / factorial(parameter.nops())).expand());
2919 lst newparameter = parameter;
2920 newparameter.remove_first();
2922 if (parameter.op(0) == 0) {
2925 ex res = convert_H_to_zeta(parameter);
2926 map_trafo_H_1mxt1px recursion;
2927 ex buffer = recursion(H(newparameter, arg).hold());
2928 if (is_a<add>(buffer)) {
2929 for (int i=0; i<buffer.nops(); i++) {
2930 res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
2933 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
2937 } else if (parameter.op(0) == -1) {
2939 // leading negative one
2940 ex res = convert_H_to_zeta(parameter);
2941 map_trafo_H_1mxt1px recursion;
2942 ex buffer = recursion(H(newparameter, arg).hold());
2943 if (is_a<add>(buffer)) {
2944 for (int i=0; i<buffer.nops(); i++) {
2945 res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
2948 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
2955 map_trafo_H_1mxt1px recursion;
2956 map_trafo_H_mult unify;
2957 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
2959 while (parameter.op(firstzero) == 1) {
2962 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
2966 newparameter.append(parameter[j+1]);
2968 newparameter.append(1);
2969 for (; j<parameter.nops()-1; j++) {
2970 newparameter.append(parameter[j+1]);
2972 res -= H(newparameter, arg).hold();
2974 res = recursion(res).expand() / firstzero;
2986 // do the actual summation.
2987 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
2989 const int j = m.size();
2991 std::vector<cln::cl_N> t(j);
2993 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2994 cln::cl_N factor = cln::expt(x, j) * one;
3000 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
3001 for (int k=j-2; k>=1; k--) {
3002 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
3004 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
3005 factor = factor * x;
3006 } while (t[0] != t0buf);
3012 } // end of anonymous namespace
3015 //////////////////////////////////////////////////////////////////////
3017 // Harmonic polylogarithm H(m,x)
3021 //////////////////////////////////////////////////////////////////////
3024 static ex H_evalf(const ex& x1, const ex& x2)
3026 if (is_a<lst>(x1)) {
3029 if (is_a<numeric>(x2)) {
3030 x = ex_to<numeric>(x2).to_cl_N();
3032 ex x2_val = x2.evalf();
3033 if (is_a<numeric>(x2_val)) {
3034 x = ex_to<numeric>(x2_val).to_cl_N();
3038 for (int i=0; i<x1.nops(); i++) {
3039 if (!x1.op(i).info(info_flags::integer)) {
3040 return H(x1, x2).hold();
3043 if (x1.nops() < 1) {
3044 return H(x1, x2).hold();
3047 const lst& morg = ex_to<lst>(x1);
3048 // remove trailing zeros ...
3049 if (*(--morg.end()) == 0) {
3050 symbol xtemp("xtemp");
3051 map_trafo_H_reduce_trailing_zeros filter;
3052 return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
3054 // ... and expand parameter notation
3055 bool has_minus_one = false;
3057 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
3059 for (ex count=*it-1; count > 0; count--) {
3063 } else if (*it <= -1) {
3064 for (ex count=*it+1; count < 0; count++) {
3068 has_minus_one = true;
3075 if (cln::abs(x) < 0.95) {
3079 if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
3080 // negative parameters -> s_lst is filled
3081 std::vector<int> m_int;
3082 std::vector<cln::cl_N> x_cln;
3083 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
3084 it_int != m_lst.end(); it_int++, it_cln++) {
3085 m_int.push_back(ex_to<numeric>(*it_int).to_int());
3086 x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
3088 x_cln.front() = x_cln.front() * x;
3089 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
3091 // only positive parameters
3093 if (m_lst.nops() == 1) {
3094 return Li(m_lst.op(0), x2).evalf();
3096 std::vector<int> m_int;
3097 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
3098 m_int.push_back(ex_to<numeric>(*it).to_int());
3100 return numeric(H_do_sum(m_int, x));
3104 symbol xtemp("xtemp");
3107 // ensure that the realpart of the argument is positive
3108 if (cln::realpart(x) < 0) {
3110 for (int i=0; i<m.nops(); i++) {
3112 m.let_op(i) = -m.op(i);
3119 if (cln::abs(x) >= 2.0) {
3120 map_trafo_H_1overx trafo;
3121 res *= trafo(H(m, xtemp));
3122 if (cln::imagpart(x) <= 0) {
3123 res = res.subs(H_polesign == -I*Pi);
3125 res = res.subs(H_polesign == I*Pi);
3127 return res.subs(xtemp == numeric(x)).evalf();
3130 // check transformations for 0.95 <= |x| < 2.0
3132 // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47
3133 if (cln::abs(x-9.53) <= 9.47) {
3135 map_trafo_H_1mxt1px trafo;
3136 res *= trafo(H(m, xtemp));
3139 if (has_minus_one) {
3140 map_trafo_H_convert_to_Li filter;
3141 return filter(H(m, numeric(x)).hold()).evalf();
3143 map_trafo_H_1mx trafo;
3144 res *= trafo(H(m, xtemp));
3147 return res.subs(xtemp == numeric(x)).evalf();
3150 return H(x1,x2).hold();
3154 static ex H_eval(const ex& m_, const ex& x)
3157 if (is_a<lst>(m_)) {
3162 if (m.nops() == 0) {
3170 if (*m.begin() > _ex1) {
3176 } else if (*m.begin() < _ex_1) {
3182 } else if (*m.begin() == _ex0) {
3189 for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
3190 if ((*it).info(info_flags::integer)) {
3201 } else if (*it < _ex_1) {
3221 } else if (step == 1) {
3233 // if some m_i is not an integer
3234 return H(m_, x).hold();
3237 if ((x == _ex1) && (*(--m.end()) != _ex0)) {
3238 return convert_H_to_zeta(m);
3244 return H(m_, x).hold();
3246 return pow(log(x), m.nops()) / factorial(m.nops());
3249 return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
3251 } else if ((step == 1) && (pos1 == _ex0)){
3256 return pow(-1, p) * S(n, p, -x);
3262 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
3263 return H(m_, x).evalf();
3265 return H(m_, x).hold();
3269 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
3272 seq.push_back(expair(H(m, x), 0));
3273 return pseries(rel, seq);
3277 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
3279 GINAC_ASSERT(deriv_param < 2);
3280 if (deriv_param == 0) {
3284 if (is_a<lst>(m_)) {
3300 return 1/(1-x) * H(m, x);
3301 } else if (mb == _ex_1) {
3302 return 1/(1+x) * H(m, x);
3309 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
3312 if (is_a<lst>(m_)) {
3317 c.s << "\\mbox{H}_{";
3318 lst::const_iterator itm = m.begin();
3321 for (; itm != m.end(); itm++) {
3331 REGISTER_FUNCTION(H,
3332 evalf_func(H_evalf).
3334 series_func(H_series).
3335 derivative_func(H_deriv).
3336 print_func<print_latex>(H_print_latex).
3337 do_not_evalf_params());
3340 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
3341 ex convert_H_to_Li(const ex& m, const ex& x)
3343 map_trafo_H_reduce_trailing_zeros filter;
3344 map_trafo_H_convert_to_Li filter2;
3346 return filter2(filter(H(m, x).hold()));
3348 return filter2(filter(H(lst(m), x).hold()));
3353 //////////////////////////////////////////////////////////////////////
3355 // Multiple zeta values zeta(x) and zeta(x,s)
3359 //////////////////////////////////////////////////////////////////////
3362 // anonymous namespace for helper functions
3366 // parameters and data for [Cra] algorithm
3367 const cln::cl_N lambda = cln::cl_N("319/320");
3370 std::vector<std::vector<cln::cl_N> > f_kj;
3371 std::vector<cln::cl_N> crB;
3372 std::vector<std::vector<cln::cl_N> > crG;
3373 std::vector<cln::cl_N> crX;
3376 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
3378 const int size = a.size();
3379 for (int n=0; n<size; n++) {
3381 for (int m=0; m<=n; m++) {
3382 c[n] = c[n] + a[m]*b[n-m];
3389 void initcX(const std::vector<int>& s)
3391 const int k = s.size();
3397 for (int i=0; i<=L2; i++) {
3398 crB.push_back(bernoulli(i).to_cl_N() / cln::factorial(i));
3403 for (int m=0; m<k-1; m++) {
3404 std::vector<cln::cl_N> crGbuf;
3407 for (int i=0; i<=L2; i++) {
3408 crGbuf.push_back(cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2));
3410 crG.push_back(crGbuf);
3415 for (int m=0; m<k-1; m++) {
3416 std::vector<cln::cl_N> Xbuf;
3417 for (int i=0; i<=L2; i++) {
3418 Xbuf.push_back(crX[i] * crG[m][i]);
3420 halfcyclic_convolute(Xbuf, crB, crX);
3426 cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk)
3428 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3429 cln::cl_N factor = cln::expt(lambda, Sqk);
3430 cln::cl_N res = factor / Sqk * crX[0] * one;
3435 factor = factor * lambda;
3437 res = res + crX[N] * factor / (N+Sqk);
3438 } while ((res != resbuf) || cln::zerop(crX[N]));
3444 void calc_f(int maxr)
3449 cln::cl_N t0, t1, t2, t3, t4;
3451 std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
3452 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3454 t0 = cln::exp(-lambda);
3456 for (k=1; k<=L1; k++) {
3459 for (j=1; j<=maxr; j++) {
3462 for (i=2; i<=j; i++) {
3466 (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
3474 cln::cl_N crandall_Z(const std::vector<int>& s)
3476 const int j = s.size();
3485 t0 = t0 + f_kj[q+j-2][s[0]-1];
3486 } while (t0 != t0buf);
3488 return t0 / cln::factorial(s[0]-1);
3491 std::vector<cln::cl_N> t(j);
3498 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
3499 for (int k=j-2; k>=1; k--) {
3500 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
3502 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
3503 } while (t[0] != t0buf);
3505 return t[0] / cln::factorial(s[0]-1);
3510 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
3512 std::vector<int> r = s;
3513 const int j = r.size();
3515 // decide on maximal size of f_kj for crandall_Z
3519 L1 = Digits * 3 + j*2;
3522 // decide on maximal size of crX for crandall_Y
3525 } else if (Digits < 86) {
3527 } else if (Digits < 192) {
3529 } else if (Digits < 394) {
3531 } else if (Digits < 808) {
3541 for (int i=0; i<j; i++) {
3550 const cln::cl_N r0factorial = cln::factorial(r[0]-1);
3552 std::vector<int> rz;
3555 for (int k=r.size()-1; k>0; k--) {
3557 rz.insert(rz.begin(), r.back());
3558 skp1buf = rz.front();
3564 for (int q=0; q<skp1buf; q++) {
3566 cln::cl_N pp1 = crandall_Y_loop(Srun+q-k);
3567 cln::cl_N pp2 = crandall_Z(rz);
3572 res = res - pp1 * pp2 / cln::factorial(q);
3574 res = res + pp1 * pp2 / cln::factorial(q);
3577 rz.front() = skp1buf;
3579 rz.insert(rz.begin(), r.back());
3583 res = (res + crandall_Y_loop(S-j)) / r0factorial + crandall_Z(rz);
3589 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
3591 const int j = r.size();
3593 // buffer for subsums
3594 std::vector<cln::cl_N> t(j);
3595 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3602 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
3603 for (int k=j-2; k>=0; k--) {
3604 t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
3606 } while (t[0] != t0buf);
3612 // does Hoelder convolution. see [BBB] (7.0)
3613 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
3615 // prepare parameters
3616 // holds Li arguments in [BBB] notation
3617 std::vector<int> s = s_;
3618 std::vector<int> m_p = m_;
3619 std::vector<int> m_q;
3620 // holds Li arguments in nested sums notation
3621 std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
3622 s_p[0] = s_p[0] * cln::cl_N("1/2");
3623 // convert notations
3625 for (int i=0; i<s_.size(); i++) {
3630 s[i] = sig * std::abs(s[i]);
3632 std::vector<cln::cl_N> s_q;
3633 cln::cl_N signum = 1;
3636 cln::cl_N res = multipleLi_do_sum(m_p, s_p);
3641 // change parameters
3642 if (s.front() > 0) {
3643 if (m_p.front() == 1) {
3644 m_p.erase(m_p.begin());
3645 s_p.erase(s_p.begin());
3646 if (s_p.size() > 0) {
3647 s_p.front() = s_p.front() * cln::cl_N("1/2");
3653 m_q.insert(m_q.begin(), 1);
3654 if (s_q.size() > 0) {
3655 s_q.front() = s_q.front() * 2;
3657 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3660 if (m_p.front() == 1) {
3661 m_p.erase(m_p.begin());
3662 cln::cl_N spbuf = s_p.front();
3663 s_p.erase(s_p.begin());
3664 if (s_p.size() > 0) {
3665 s_p.front() = s_p.front() * spbuf;
3668 m_q.insert(m_q.begin(), 1);
3669 if (s_q.size() > 0) {
3670 s_q.front() = s_q.front() * 4;
3672 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
3676 m_q.insert(m_q.begin(), 1);
3677 if (s_q.size() > 0) {
3678 s_q.front() = s_q.front() * 2;
3680 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3685 if (m_p.size() == 0) break;
3687 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
3692 res = res + signum * multipleLi_do_sum(m_q, s_q);
3698 } // end of anonymous namespace
3701 //////////////////////////////////////////////////////////////////////
3703 // Multiple zeta values zeta(x)
3707 //////////////////////////////////////////////////////////////////////
3710 static ex zeta1_evalf(const ex& x)
3712 if (is_exactly_a<lst>(x) && (x.nops()>1)) {
3714 // multiple zeta value
3715 const int count = x.nops();
3716 const lst& xlst = ex_to<lst>(x);
3717 std::vector<int> r(count);
3719 // check parameters and convert them
3720 lst::const_iterator it1 = xlst.begin();
3721 std::vector<int>::iterator it2 = r.begin();
3723 if (!(*it1).info(info_flags::posint)) {
3724 return zeta(x).hold();
3726 *it2 = ex_to<numeric>(*it1).to_int();
3729 } while (it2 != r.end());
3731 // check for divergence
3733 return zeta(x).hold();
3736 // decide on summation algorithm
3737 // this is still a bit clumsy
3738 int limit = (Digits>17) ? 10 : 6;
3739 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
3740 return numeric(zeta_do_sum_Crandall(r));
3742 return numeric(zeta_do_sum_simple(r));
3746 // single zeta value
3747 if (is_exactly_a<numeric>(x) && (x != 1)) {
3749 return zeta(ex_to<numeric>(x));
3750 } catch (const dunno &e) { }
3753 return zeta(x).hold();
3757 static ex zeta1_eval(const ex& m)
3759 if (is_exactly_a<lst>(m)) {
3760 if (m.nops() == 1) {
3761 return zeta(m.op(0));
3763 return zeta(m).hold();
3766 if (m.info(info_flags::numeric)) {
3767 const numeric& y = ex_to<numeric>(m);
3768 // trap integer arguments:
3769 if (y.is_integer()) {
3773 if (y.is_equal(*_num1_p)) {
3774 return zeta(m).hold();
3776 if (y.info(info_flags::posint)) {
3777 if (y.info(info_flags::odd)) {
3778 return zeta(m).hold();
3780 return abs(bernoulli(y)) * pow(Pi, y) * pow(*_num2_p, y-(*_num1_p)) / factorial(y);
3783 if (y.info(info_flags::odd)) {
3784 return -bernoulli((*_num1_p)-y) / ((*_num1_p)-y);
3791 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
3792 return zeta1_evalf(m);
3795 return zeta(m).hold();
3799 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
3801 GINAC_ASSERT(deriv_param==0);
3803 if (is_exactly_a<lst>(m)) {
3806 return zetaderiv(_ex1, m);
3811 static void zeta1_print_latex(const ex& m_, const print_context& c)
3814 if (is_a<lst>(m_)) {
3815 const lst& m = ex_to<lst>(m_);
3816 lst::const_iterator it = m.begin();
3819 for (; it != m.end(); it++) {
3830 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1).
3831 evalf_func(zeta1_evalf).
3832 eval_func(zeta1_eval).
3833 derivative_func(zeta1_deriv).
3834 print_func<print_latex>(zeta1_print_latex).
3835 do_not_evalf_params().
3839 //////////////////////////////////////////////////////////////////////
3841 // Alternating Euler sum zeta(x,s)
3845 //////////////////////////////////////////////////////////////////////
3848 static ex zeta2_evalf(const ex& x, const ex& s)
3850 if (is_exactly_a<lst>(x)) {
3852 // alternating Euler sum
3853 const int count = x.nops();
3854 const lst& xlst = ex_to<lst>(x);
3855 const lst& slst = ex_to<lst>(s);
3856 std::vector<int> xi(count);
3857 std::vector<int> si(count);
3859 // check parameters and convert them
3860 lst::const_iterator it_xread = xlst.begin();
3861 lst::const_iterator it_sread = slst.begin();
3862 std::vector<int>::iterator it_xwrite = xi.begin();
3863 std::vector<int>::iterator it_swrite = si.begin();
3865 if (!(*it_xread).info(info_flags::posint)) {
3866 return zeta(x, s).hold();
3868 *it_xwrite = ex_to<numeric>(*it_xread).to_int();
3869 if (*it_sread > 0) {
3878 } while (it_xwrite != xi.end());
3880 // check for divergence
3881 if ((xi[0] == 1) && (si[0] == 1)) {
3882 return zeta(x, s).hold();
3885 // use Hoelder convolution
3886 return numeric(zeta_do_Hoelder_convolution(xi, si));
3889 return zeta(x, s).hold();
3893 static ex zeta2_eval(const ex& m, const ex& s_)
3895 if (is_exactly_a<lst>(s_)) {
3896 const lst& s = ex_to<lst>(s_);
3897 for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
3898 if ((*it).info(info_flags::positive)) {
3901 return zeta(m, s_).hold();
3904 } else if (s_.info(info_flags::positive)) {
3908 return zeta(m, s_).hold();
3912 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
3914 GINAC_ASSERT(deriv_param==0);
3916 if (is_exactly_a<lst>(m)) {
3919 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
3920 return zetaderiv(_ex1, m);
3927 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
3930 if (is_a<lst>(m_)) {
3936 if (is_a<lst>(s_)) {
3942 lst::const_iterator itm = m.begin();
3943 lst::const_iterator its = s.begin();
3945 c.s << "\\overline{";
3953 for (; itm != m.end(); itm++, its++) {
3956 c.s << "\\overline{";
3967 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2).
3968 evalf_func(zeta2_evalf).
3969 eval_func(zeta2_eval).
3970 derivative_func(zeta2_deriv).
3971 print_func<print_latex>(zeta2_print_latex).
3972 do_not_evalf_params().
3976 } // namespace GiNaC