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 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]);
496 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
497 for (int k=j-2; k>=0; k--) {
498 flag_accidental_zero = cln::zerop(t[k+1]);
499 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]);
501 } while ( (t[0] != t0buf) || flag_accidental_zero );
507 // converts parameter types and calls multipleLi_do_sum (convenience function for G_numeric)
508 cln::cl_N mLi_do_summation(const lst& m, const lst& x)
510 std::vector<int> m_int;
511 std::vector<cln::cl_N> x_cln;
512 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
513 m_int.push_back(ex_to<numeric>(*itm).to_int());
514 x_cln.push_back(ex_to<numeric>(*itx).to_cl_N());
516 return multipleLi_do_sum(m_int, x_cln);
520 // forward declaration for Li_eval()
521 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
524 // holding dummy-symbols for the G/Li transformations
525 std::vector<ex> gsyms;
528 // type used by the transformation functions for G
529 typedef std::vector<int> Gparameter;
532 // G_eval1-function for G transformations
533 ex G_eval1(int a, int scale)
536 const ex& scs = gsyms[std::abs(scale)];
537 const ex& as = gsyms[std::abs(a)];
539 return -log(1 - scs/as);
544 return log(gsyms[std::abs(scale)]);
549 // G_eval-function for G transformations
550 ex G_eval(const Gparameter& a, int scale)
552 // check for properties of G
553 ex sc = gsyms[std::abs(scale)];
555 bool all_zero = true;
556 bool all_ones = true;
558 for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
560 const ex sym = gsyms[std::abs(*it)];
574 // care about divergent G: shuffle to separate divergencies that will be canceled
575 // later on in the transformation
576 if (newa.nops() > 1 && newa.op(0) == sc && !all_ones && a.front()!=0) {
579 Gparameter::const_iterator it = a.begin();
581 for (; it != a.end(); ++it) {
582 short_a.push_back(*it);
584 ex result = G_eval1(a.front(), scale) * G_eval(short_a, scale);
585 it = short_a.begin();
586 for (int i=1; i<count_ones; ++i) {
589 for (; it != short_a.end(); ++it) {
592 Gparameter::const_iterator it2 = short_a.begin();
593 for (--it2; it2 != it;) {
595 newa.push_back(*it2);
597 newa.push_back(a[0]);
599 for (; it2 != short_a.end(); ++it2) {
600 newa.push_back(*it2);
602 result -= G_eval(newa, scale);
604 return result / count_ones;
607 // G({1,...,1};y) -> G({1};y)^k / k!
608 if (all_ones && a.size() > 1) {
609 return pow(G_eval1(a.front(),scale), count_ones) / factorial(count_ones);
612 // G({0,...,0};y) -> log(y)^k / k!
614 return pow(log(gsyms[std::abs(scale)]), a.size()) / factorial(a.size());
617 // no special cases anymore -> convert it into Li
620 ex argbuf = gsyms[std::abs(scale)];
622 for (Gparameter::const_iterator it=a.begin(); it!=a.end(); ++it) {
624 const ex& sym = gsyms[std::abs(*it)];
625 x.append(argbuf / sym);
633 return pow(-1, x.nops()) * Li(m, x);
637 // converts data for G: pending_integrals -> a
638 Gparameter convert_pending_integrals_G(const Gparameter& pending_integrals)
640 GINAC_ASSERT(pending_integrals.size() != 1);
642 if (pending_integrals.size() > 0) {
643 // get rid of the first element, which would stand for the new upper limit
644 Gparameter new_a(pending_integrals.begin()+1, pending_integrals.end());
647 // just return empty parameter list
654 // check the parameters a and scale for G and return information about convergence, depth, etc.
655 // convergent : true if G(a,scale) is convergent
656 // depth : depth of G(a,scale)
657 // trailing_zeros : number of trailing zeros of a
658 // min_it : iterator of a pointing on the smallest element in a
659 Gparameter::const_iterator check_parameter_G(const Gparameter& a, int scale,
660 bool& convergent, int& depth, int& trailing_zeros, Gparameter::const_iterator& min_it)
666 Gparameter::const_iterator lastnonzero = a.end();
667 for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
668 if (std::abs(*it) > 0) {
672 if (std::abs(*it) < scale) {
674 if ((min_it == a.end()) || (std::abs(*it) < std::abs(*min_it))) {
682 return ++lastnonzero;
686 // add scale to pending_integrals if pending_integrals is empty
687 Gparameter prepare_pending_integrals(const Gparameter& pending_integrals, int scale)
689 GINAC_ASSERT(pending_integrals.size() != 1);
691 if (pending_integrals.size() > 0) {
692 return pending_integrals;
694 Gparameter new_pending_integrals;
695 new_pending_integrals.push_back(scale);
696 return new_pending_integrals;
701 // handles trailing zeroes for an otherwise convergent integral
702 ex trailing_zeros_G(const Gparameter& a, int scale)
705 int depth, trailing_zeros;
706 Gparameter::const_iterator last, dummyit;
707 last = check_parameter_G(a, scale, convergent, depth, trailing_zeros, dummyit);
709 GINAC_ASSERT(convergent);
711 if ((trailing_zeros > 0) && (depth > 0)) {
713 Gparameter new_a(a.begin(), a.end()-1);
714 result += G_eval1(0, scale) * trailing_zeros_G(new_a, scale);
715 for (Gparameter::const_iterator it = a.begin(); it != last; ++it) {
716 Gparameter new_a(a.begin(), it);
718 new_a.insert(new_a.end(), it, a.end()-1);
719 result -= trailing_zeros_G(new_a, scale);
722 return result / trailing_zeros;
724 return G_eval(a, scale);
729 // G transformation [VSW] (57),(58)
730 ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, int scale)
732 // pendint = ( y1, b1, ..., br )
733 // a = ( 0, ..., 0, amin )
736 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(0, ..., 0, sr; y2)
737 // where sr replaces amin
739 GINAC_ASSERT(a.back() != 0);
740 GINAC_ASSERT(a.size() > 0);
743 Gparameter new_pending_integrals = prepare_pending_integrals(pending_integrals, std::abs(a.back()));
744 const int psize = pending_integrals.size();
747 // G(sr_{+-}; y2 ) = G(y2_{-+}; sr) - G(0; sr) + ln(-y2_{-+})
752 result += log(gsyms[ex_to<numeric>(scale).to_int()]);
754 new_pending_integrals.push_back(-scale);
757 new_pending_integrals.push_back(scale);
761 result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals), pending_integrals.front());
765 result += trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals), new_pending_integrals.front());
768 new_pending_integrals.back() = 0;
769 result -= trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals), new_pending_integrals.front());
775 // G_m(sr_{+-}; y2) = -zeta_m + int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
776 // - int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
779 result -= zeta(a.size());
781 result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals), pending_integrals.front());
784 // term int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
785 // = int_0^sr dt/t G_{m-1}( t_{+-}; y2 )
786 Gparameter new_a(a.begin()+1, a.end());
787 new_pending_integrals.push_back(0);
788 result -= depth_one_trafo_G(new_pending_integrals, new_a, scale);
790 // term int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
791 // = int_0^y2 dt/t G_{m-1}( t_{+-}; y2 )
792 Gparameter new_pending_integrals_2;
793 new_pending_integrals_2.push_back(scale);
794 new_pending_integrals_2.push_back(0);
796 result += trailing_zeros_G(convert_pending_integrals_G(pending_integrals), pending_integrals.front())
797 * depth_one_trafo_G(new_pending_integrals_2, new_a, scale);
799 result += depth_one_trafo_G(new_pending_integrals_2, new_a, scale);
806 // forward declaration
807 ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
808 const Gparameter& pendint, const Gparameter& a_old, int scale);
811 // G transformation [VSW]
812 ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale)
814 // main recursion routine
816 // pendint = ( y1, b1, ..., br )
817 // a = ( a1, ..., amin, ..., aw )
820 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
821 // where sr replaces amin
823 // find smallest alpha, determine depth and trailing zeros, and check for convergence
825 int depth, trailing_zeros;
826 Gparameter::const_iterator min_it;
827 Gparameter::const_iterator firstzero =
828 check_parameter_G(a, scale, convergent, depth, trailing_zeros, min_it);
829 int min_it_pos = min_it - a.begin();
831 // special case: all a's are zero
838 result = G_eval(a, scale);
840 if (pendint.size() > 0) {
841 result *= trailing_zeros_G(convert_pending_integrals_G(pendint), pendint.front());
846 // handle trailing zeros
847 if (trailing_zeros > 0) {
849 Gparameter new_a(a.begin(), a.end()-1);
850 result += G_eval1(0, scale) * G_transform(pendint, new_a, scale);
851 for (Gparameter::const_iterator it = a.begin(); it != firstzero; ++it) {
852 Gparameter new_a(a.begin(), it);
854 new_a.insert(new_a.end(), it, a.end()-1);
855 result -= G_transform(pendint, new_a, scale);
857 return result / trailing_zeros;
862 if (pendint.size() > 0) {
863 return G_eval(convert_pending_integrals_G(pendint), pendint.front()) * G_eval(a, scale);
865 return G_eval(a, scale);
869 // call basic transformation for depth equal one
871 return depth_one_trafo_G(pendint, a, scale);
875 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
876 // = int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,0,...,aw,y2)
877 // + 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)
879 // smallest element in last place
880 if (min_it + 1 == a.end()) {
881 do { --min_it; } while (*min_it == 0);
883 Gparameter a1(a.begin(),min_it+1);
884 Gparameter a2(min_it+1,a.end());
886 ex result = G_transform(pendint,a2,scale)*G_transform(empty,a1,scale);
888 result -= shuffle_G(empty,a1,a2,pendint,a,scale);
893 Gparameter::iterator changeit;
895 // first term G(a_1,..,0,...,a_w;a_0)
896 Gparameter new_pendint = prepare_pending_integrals(pendint, a[min_it_pos]);
897 Gparameter new_a = a;
898 new_a[min_it_pos] = 0;
899 ex result = G_transform(empty, new_a, scale);
900 if (pendint.size() > 0) {
901 result *= trailing_zeros_G(convert_pending_integrals_G(pendint), pendint.front());
905 changeit = new_a.begin() + min_it_pos;
906 changeit = new_a.erase(changeit);
907 if (changeit != new_a.begin()) {
908 // smallest in the middle
909 new_pendint.push_back(*changeit);
910 result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
911 * G_transform(empty, new_a, scale);
912 int buffer = *changeit;
914 result += G_transform(new_pendint, new_a, scale);
916 new_pendint.pop_back();
918 new_pendint.push_back(*changeit);
919 result += trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
920 * G_transform(empty, new_a, scale);
922 result -= G_transform(new_pendint, new_a, scale);
924 // smallest at the front
925 new_pendint.push_back(scale);
926 result += trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
927 * G_transform(empty, new_a, scale);
928 new_pendint.back() = *changeit;
929 result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
930 * G_transform(empty, new_a, scale);
932 result += G_transform(new_pendint, new_a, scale);
938 // shuffles the two parameter list a1 and a2 and calls G_transform for every term except
939 // for the one that is equal to a_old
940 ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
941 const Gparameter& pendint, const Gparameter& a_old, int scale)
943 if (a1.size()==0 && a2.size()==0) {
944 // veto the one configuration we don't want
945 if ( a0 == a_old ) return 0;
947 return G_transform(pendint,a0,scale);
953 aa0.insert(aa0.end(),a1.begin(),a1.end());
954 return shuffle_G(aa0,empty,empty,pendint,a_old,scale);
960 aa0.insert(aa0.end(),a2.begin(),a2.end());
961 return shuffle_G(aa0,empty,empty,pendint,a_old,scale);
964 Gparameter a1_removed(a1.begin()+1,a1.end());
965 Gparameter a2_removed(a2.begin()+1,a2.end());
970 a01.push_back( a1[0] );
971 a02.push_back( a2[0] );
973 return shuffle_G(a01,a1_removed,a2,pendint,a_old,scale)
974 + shuffle_G(a02,a1,a2_removed,pendint,a_old,scale);
978 // handles the transformations and the numerical evaluation of G
979 // the parameter x, s and y must only contain numerics
980 ex G_numeric(const lst& x, const lst& s, const ex& y)
982 // check for convergence and necessary accelerations
983 bool need_trafo = false;
984 bool need_hoelder = false;
986 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
987 if (!(*it).is_zero()) {
989 if (abs(*it) - y < -pow(10,-Digits+1)) {
992 if (abs((abs(*it) - y)/y) < 0.01) {
997 if (x.op(x.nops()-1).is_zero()) {
1000 if (depth == 1 && x.nops() == 2 && !need_trafo) {
1001 return -Li(x.nops(), y / x.op(x.nops()-1)).evalf();
1004 // do acceleration transformation (hoelder convolution [BBB])
1008 const int size = x.nops();
1010 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1011 newx.append(*it / y);
1014 for (int r=0; r<=size; ++r) {
1015 ex buffer = pow(-1, r);
1020 for (lst::const_iterator it = newx.begin(); it != newx.end(); ++it) {
1031 for (int j=r; j>=1; --j) {
1032 qlstx.append(1-newx.op(j-1));
1033 if (newx.op(j-1).info(info_flags::real) && newx.op(j-1) > 1 && newx.op(j-1) <= 2) {
1034 qlsts.append( s.op(j-1));
1036 qlsts.append( -s.op(j-1));
1039 if (qlstx.nops() > 0) {
1040 buffer *= G_numeric(qlstx, qlsts, 1/q);
1044 for (int j=r+1; j<=size; ++j) {
1045 plstx.append(newx.op(j-1));
1046 plsts.append(s.op(j-1));
1048 if (plstx.nops() > 0) {
1049 buffer *= G_numeric(plstx, plsts, 1/p);
1056 // convergence transformation
1059 // sort (|x|<->position) to determine indices
1060 std::multimap<ex,int> sortmap;
1062 for (int i=0; i<x.nops(); ++i) {
1063 if (!x[i].is_zero()) {
1064 sortmap.insert(std::pair<ex,int>(abs(x[i]), i));
1068 // include upper limit (scale)
1069 sortmap.insert(std::pair<ex,int>(abs(y), x.nops()));
1071 // generate missing dummy-symbols
1074 gsyms.push_back(symbol("GSYMS_ERROR"));
1076 for (std::multimap<ex,int>::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1077 if (it != sortmap.begin()) {
1078 if (it->second < x.nops()) {
1079 if (x[it->second] == lastentry) {
1080 gsyms.push_back(gsyms.back());
1084 if (y == lastentry) {
1085 gsyms.push_back(gsyms.back());
1090 std::ostringstream os;
1092 gsyms.push_back(symbol(os.str()));
1094 if (it->second < x.nops()) {
1095 lastentry = x[it->second];
1101 // fill position data according to sorted indices and prepare substitution list
1102 Gparameter a(x.nops());
1106 for (std::multimap<ex,int>::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1107 if (it->second < x.nops()) {
1108 if (s[it->second] > 0) {
1109 a[it->second] = pos;
1111 a[it->second] = -pos;
1113 subslst.append(gsyms[pos] == x[it->second]);
1116 subslst.append(gsyms[pos] == y);
1121 // do transformation
1123 ex result = G_transform(pendint, a, scale);
1124 // replace dummy symbols with their values
1125 result = result.eval().expand();
1126 result = result.subs(subslst).evalf();
1137 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1138 if ((*it).is_zero()) {
1141 newx.append(factor / (*it));
1149 return sign * numeric(mLi_do_summation(m, newx));
1153 ex mLi_numeric(const lst& m, const lst& x)
1155 // let G_numeric do the transformation
1159 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1160 for (int i = 1; i < *itm; ++i) {
1164 newx.append(factor / *itx);
1168 return pow(-1, m.nops()) * G_numeric(newx, s, _ex1);
1172 } // end of anonymous namespace
1175 //////////////////////////////////////////////////////////////////////
1177 // Generalized multiple polylogarithm G(x, y) and G(x, s, y)
1181 //////////////////////////////////////////////////////////////////////
1184 static ex G2_evalf(const ex& x_, const ex& y)
1186 if (!y.info(info_flags::positive)) {
1187 return G(x_, y).hold();
1189 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1190 if (x.nops() == 0) {
1194 return G(x_, y).hold();
1197 bool all_zero = true;
1198 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1199 if (!(*it).info(info_flags::numeric)) {
1200 return G(x_, y).hold();
1208 return pow(log(y), x.nops()) / factorial(x.nops());
1210 return G_numeric(x, s, y);
1214 static ex G2_eval(const ex& x_, const ex& y)
1216 //TODO eval to MZV or H or S or Lin
1218 if (!y.info(info_flags::positive)) {
1219 return G(x_, y).hold();
1221 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1222 if (x.nops() == 0) {
1226 return G(x_, y).hold();
1229 bool all_zero = true;
1230 bool crational = true;
1231 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1232 if (!(*it).info(info_flags::numeric)) {
1233 return G(x_, y).hold();
1235 if (!(*it).info(info_flags::crational)) {
1244 return pow(log(y), x.nops()) / factorial(x.nops());
1246 if (!y.info(info_flags::crational)) {
1250 return G(x_, y).hold();
1252 return G_numeric(x, s, y);
1256 unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2).
1257 evalf_func(G2_evalf).
1259 do_not_evalf_params().
1262 // derivative_func(G2_deriv).
1263 // print_func<print_latex>(G2_print_latex).
1266 static ex G3_evalf(const ex& x_, const ex& s_, const ex& y)
1268 if (!y.info(info_flags::positive)) {
1269 return G(x_, s_, y).hold();
1271 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1272 lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
1273 if (x.nops() != s.nops()) {
1274 return G(x_, s_, y).hold();
1276 if (x.nops() == 0) {
1280 return G(x_, s_, y).hold();
1283 bool all_zero = true;
1284 for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1285 if (!(*itx).info(info_flags::numeric)) {
1286 return G(x_, y).hold();
1288 if (!(*its).info(info_flags::real)) {
1289 return G(x_, y).hold();
1301 return pow(log(y), x.nops()) / factorial(x.nops());
1303 return G_numeric(x, sn, y);
1307 static ex G3_eval(const ex& x_, const ex& s_, const ex& y)
1309 //TODO eval to MZV or H or S or Lin
1311 if (!y.info(info_flags::positive)) {
1312 return G(x_, s_, y).hold();
1314 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1315 lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
1316 if (x.nops() != s.nops()) {
1317 return G(x_, s_, y).hold();
1319 if (x.nops() == 0) {
1323 return G(x_, s_, y).hold();
1326 bool all_zero = true;
1327 bool crational = true;
1328 for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1329 if (!(*itx).info(info_flags::numeric)) {
1330 return G(x_, s_, y).hold();
1332 if (!(*its).info(info_flags::real)) {
1333 return G(x_, s_, y).hold();
1335 if (!(*itx).info(info_flags::crational)) {
1348 return pow(log(y), x.nops()) / factorial(x.nops());
1350 if (!y.info(info_flags::crational)) {
1354 return G(x_, s_, y).hold();
1356 return G_numeric(x, sn, y);
1360 unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3).
1361 evalf_func(G3_evalf).
1363 do_not_evalf_params().
1366 // derivative_func(G3_deriv).
1367 // print_func<print_latex>(G3_print_latex).
1370 //////////////////////////////////////////////////////////////////////
1372 // Classical polylogarithm and multiple polylogarithm Li(m,x)
1376 //////////////////////////////////////////////////////////////////////
1379 static ex Li_evalf(const ex& m_, const ex& x_)
1381 // classical polylogs
1382 if (m_.info(info_flags::posint)) {
1383 if (x_.info(info_flags::numeric)) {
1384 return Lin_numeric(ex_to<numeric>(m_).to_int(), ex_to<numeric>(x_));
1386 // try to numerically evaluate second argument
1387 ex x_val = x_.evalf();
1388 if (x_val.info(info_flags::numeric)) {
1389 return Lin_numeric(ex_to<numeric>(m_).to_int(), ex_to<numeric>(x_val));
1393 // multiple polylogs
1394 if (is_a<lst>(m_) && is_a<lst>(x_)) {
1396 const lst& m = ex_to<lst>(m_);
1397 const lst& x = ex_to<lst>(x_);
1398 if (m.nops() != x.nops()) {
1399 return Li(m_,x_).hold();
1401 if (x.nops() == 0) {
1404 if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) {
1405 return Li(m_,x_).hold();
1408 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1409 if (!(*itm).info(info_flags::posint)) {
1410 return Li(m_, x_).hold();
1412 if (!(*itx).info(info_flags::numeric)) {
1413 return Li(m_, x_).hold();
1420 return mLi_numeric(m, x);
1423 return Li(m_,x_).hold();
1427 static ex Li_eval(const ex& m_, const ex& x_)
1429 if (is_a<lst>(m_)) {
1430 if (is_a<lst>(x_)) {
1431 // multiple polylogs
1432 const lst& m = ex_to<lst>(m_);
1433 const lst& x = ex_to<lst>(x_);
1434 if (m.nops() != x.nops()) {
1435 return Li(m_,x_).hold();
1437 if (x.nops() == 0) {
1441 bool is_zeta = true;
1442 bool do_evalf = true;
1443 bool crational = true;
1444 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1445 if (!(*itm).info(info_flags::posint)) {
1446 return Li(m_,x_).hold();
1448 if ((*itx != _ex1) && (*itx != _ex_1)) {
1449 if (itx != x.begin()) {
1457 if (!(*itx).info(info_flags::numeric)) {
1460 if (!(*itx).info(info_flags::crational)) {
1469 lst newm = convert_parameter_Li_to_H(m, x, prefactor);
1470 return prefactor * H(newm, x[0]);
1472 if (do_evalf && !crational) {
1473 return mLi_numeric(m,x);
1476 return Li(m_, x_).hold();
1477 } else if (is_a<lst>(x_)) {
1478 return Li(m_, x_).hold();
1481 // classical polylogs
1489 return (pow(2,1-m_)-1) * zeta(m_);
1495 if (x_.is_equal(I)) {
1496 return power(Pi,_ex2)/_ex_48 + Catalan*I;
1498 if (x_.is_equal(-I)) {
1499 return power(Pi,_ex2)/_ex_48 - Catalan*I;
1502 if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
1503 return Lin_numeric(ex_to<numeric>(m_).to_int(), ex_to<numeric>(x_));
1506 return Li(m_, x_).hold();
1510 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
1512 if (is_a<lst>(m) || is_a<lst>(x)) {
1515 seq.push_back(expair(Li(m, x), 0));
1516 return pseries(rel, seq);
1519 // classical polylog
1520 const ex x_pt = x.subs(rel, subs_options::no_pattern);
1521 if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
1522 // First special case: x==0 (derivatives have poles)
1523 if (x_pt.is_zero()) {
1526 // manually construct the primitive expansion
1527 for (int i=1; i<order; ++i)
1528 ser += pow(s,i) / pow(numeric(i), m);
1529 // substitute the argument's series expansion
1530 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
1531 // maybe that was terminating, so add a proper order term
1533 nseq.push_back(expair(Order(_ex1), order));
1534 ser += pseries(rel, nseq);
1535 // reexpanding it will collapse the series again
1536 return ser.series(rel, order);
1538 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
1539 throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
1541 // all other cases should be safe, by now:
1542 throw do_taylor(); // caught by function::series()
1546 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
1548 GINAC_ASSERT(deriv_param < 2);
1549 if (deriv_param == 0) {
1552 if (m_.nops() > 1) {
1553 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
1556 if (is_a<lst>(m_)) {
1562 if (is_a<lst>(x_)) {
1568 return Li(m-1, x) / x;
1575 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
1578 if (is_a<lst>(m_)) {
1584 if (is_a<lst>(x_)) {
1589 c.s << "\\mbox{Li}_{";
1590 lst::const_iterator itm = m.begin();
1593 for (; itm != m.end(); itm++) {
1598 lst::const_iterator itx = x.begin();
1601 for (; itx != x.end(); itx++) {
1609 REGISTER_FUNCTION(Li,
1610 evalf_func(Li_evalf).
1612 series_func(Li_series).
1613 derivative_func(Li_deriv).
1614 print_func<print_latex>(Li_print_latex).
1615 do_not_evalf_params());
1618 //////////////////////////////////////////////////////////////////////
1620 // Nielsen's generalized polylogarithm S(n,p,x)
1624 //////////////////////////////////////////////////////////////////////
1627 // anonymous namespace for helper functions
1631 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
1633 std::vector<std::vector<cln::cl_N> > Yn;
1634 int ynsize = 0; // number of Yn[]
1635 int ynlength = 100; // initial length of all Yn[i]
1638 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
1639 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
1640 // representing S_{n,p}(x).
1641 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
1642 // equivalent Z-sum.
1643 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
1644 // representing S_{n,p}(x).
1645 // The calculation of Y_n uses the values from Y_{n-1}.
1646 void fill_Yn(int n, const cln::float_format_t& prec)
1648 const int initsize = ynlength;
1649 //const int initsize = initsize_Yn;
1650 cln::cl_N one = cln::cl_float(1, prec);
1653 std::vector<cln::cl_N> buf(initsize);
1654 std::vector<cln::cl_N>::iterator it = buf.begin();
1655 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1656 *it = (*itprev) / cln::cl_N(n+1) * one;
1659 // sums with an index smaller than the depth are zero and need not to be calculated.
1660 // calculation starts with depth, which is n+2)
1661 for (int i=n+2; i<=initsize+n; i++) {
1662 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1668 std::vector<cln::cl_N> buf(initsize);
1669 std::vector<cln::cl_N>::iterator it = buf.begin();
1672 for (int i=2; i<=initsize; i++) {
1673 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1682 // make Yn longer ...
1683 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
1686 cln::cl_N one = cln::cl_float(1, prec);
1688 Yn[0].resize(newsize);
1689 std::vector<cln::cl_N>::iterator it = Yn[0].begin();
1691 for (int i=ynlength+1; i<=newsize; i++) {
1692 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1696 for (int n=1; n<ynsize; n++) {
1697 Yn[n].resize(newsize);
1698 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
1699 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1702 for (int i=ynlength+n+1; i<=newsize+n; i++) {
1703 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1713 // helper function for S(n,p,x)
1715 cln::cl_N C(int n, int p)
1719 for (int k=0; k<p; k++) {
1720 for (int j=0; j<=(n+k-1)/2; j++) {
1724 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
1727 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
1734 result = result + cln::factorial(n+k-1)
1735 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
1736 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1739 result = result - cln::factorial(n+k-1)
1740 * 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));
1746 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()
1747 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1750 result = result + cln::factorial(n+k-1)
1751 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
1752 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1760 if (((np)/2+n) & 1) {
1761 result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1764 result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1772 // helper function for S(n,p,x)
1773 // [Kol] remark to (9.1)
1774 cln::cl_N a_k(int k)
1783 for (int m=2; m<=k; m++) {
1784 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
1791 // helper function for S(n,p,x)
1792 // [Kol] remark to (9.1)
1793 cln::cl_N b_k(int k)
1802 for (int m=2; m<=k; m++) {
1803 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
1810 // helper function for S(n,p,x)
1811 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1814 return Li_projection(n+1, x, prec);
1817 // check if precalculated values are sufficient
1819 for (int i=ynsize; i<p-1; i++) {
1824 // should be done otherwise
1825 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1826 cln::cl_N xf = x * one;
1827 //cln::cl_N xf = x * cln::cl_float(1, prec);
1831 cln::cl_N factor = cln::expt(xf, p);
1835 if (i-p >= ynlength) {
1837 make_Yn_longer(ynlength*2, prec);
1839 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? ...
1840 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
1841 factor = factor * xf;
1843 } while (res != resbuf);
1849 // helper function for S(n,p,x)
1850 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1853 if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
1855 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
1856 * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
1858 for (int s=0; s<n; s++) {
1860 for (int r=0; r<p; r++) {
1861 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
1862 * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
1864 result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
1870 return S_do_sum(n, p, x, prec);
1874 // helper function for S(n,p,x)
1875 numeric S_num(int n, int p, const numeric& x)
1879 // [Kol] (2.22) with (2.21)
1880 return cln::zeta(p+1);
1885 return cln::zeta(n+1);
1890 for (int nu=0; nu<n; nu++) {
1891 for (int rho=0; rho<=p; rho++) {
1892 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
1893 * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
1896 result = result * cln::expt(cln::cl_I(-1),n+p-1);
1903 return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
1905 // throw std::runtime_error("don't know how to evaluate this function!");
1908 // what is the desired float format?
1909 // first guess: default format
1910 cln::float_format_t prec = cln::default_float_format;
1911 const cln::cl_N value = x.to_cl_N();
1912 // second guess: the argument's format
1913 if (!x.real().is_rational())
1914 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
1915 else if (!x.imag().is_rational())
1916 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
1919 if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95))) {
1921 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
1922 * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
1924 for (int s=0; s<n; s++) {
1926 for (int r=0; r<p; r++) {
1927 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
1928 * S_num(p-r,n-s,1-value).to_cl_N() / cln::factorial(r);
1930 result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
1937 if (cln::abs(value) > 1) {
1941 for (int s=0; s<p; s++) {
1942 for (int r=0; r<=s; r++) {
1943 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
1944 / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
1945 * S_num(n+s-r,p-s,cln::recip(value)).to_cl_N();
1948 result = result * cln::expt(cln::cl_I(-1),n);
1951 for (int r=0; r<n; r++) {
1952 res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
1954 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
1956 result = result + cln::expt(cln::cl_I(-1),p) * res2;
1961 return S_projection(n, p, value, prec);
1966 } // end of anonymous namespace
1969 //////////////////////////////////////////////////////////////////////
1971 // Nielsen's generalized polylogarithm S(n,p,x)
1975 //////////////////////////////////////////////////////////////////////
1978 static ex S_evalf(const ex& n, const ex& p, const ex& x)
1980 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
1981 if (is_a<numeric>(x)) {
1982 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1984 ex x_val = x.evalf();
1985 if (is_a<numeric>(x_val)) {
1986 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x_val));
1990 return S(n, p, x).hold();
1994 static ex S_eval(const ex& n, const ex& p, const ex& x)
1996 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2002 for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
2010 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2011 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
2016 return pow(-log(1-x), p) / factorial(p);
2018 return S(n, p, x).hold();
2022 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
2025 return Li(n+1, x).series(rel, order, options);
2028 const ex x_pt = x.subs(rel, subs_options::no_pattern);
2029 if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
2030 // First special case: x==0 (derivatives have poles)
2031 if (x_pt.is_zero()) {
2034 // manually construct the primitive expansion
2035 // subsum = Euler-Zagier-Sum is needed
2036 // dirty hack (slow ...) calculation of subsum:
2037 std::vector<ex> presubsum, subsum;
2038 subsum.push_back(0);
2039 for (int i=1; i<order-1; ++i) {
2040 subsum.push_back(subsum[i-1] + numeric(1, i));
2042 for (int depth=2; depth<p; ++depth) {
2044 for (int i=1; i<order-1; ++i) {
2045 subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
2049 for (int i=1; i<order; ++i) {
2050 ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
2052 // substitute the argument's series expansion
2053 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
2054 // maybe that was terminating, so add a proper order term
2056 nseq.push_back(expair(Order(_ex1), order));
2057 ser += pseries(rel, nseq);
2058 // reexpanding it will collapse the series again
2059 return ser.series(rel, order);
2061 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
2062 throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
2064 // all other cases should be safe, by now:
2065 throw do_taylor(); // caught by function::series()
2069 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
2071 GINAC_ASSERT(deriv_param < 3);
2072 if (deriv_param < 2) {
2076 return S(n-1, p, x) / x;
2078 return S(n, p-1, x) / (1-x);
2083 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
2085 c.s << "\\mbox{S}_{";
2095 REGISTER_FUNCTION(S,
2096 evalf_func(S_evalf).
2098 series_func(S_series).
2099 derivative_func(S_deriv).
2100 print_func<print_latex>(S_print_latex).
2101 do_not_evalf_params());
2104 //////////////////////////////////////////////////////////////////////
2106 // Harmonic polylogarithm H(m,x)
2110 //////////////////////////////////////////////////////////////////////
2113 // anonymous namespace for helper functions
2117 // regulates the pole (used by 1/x-transformation)
2118 symbol H_polesign("IMSIGN");
2121 // convert parameters from H to Li representation
2122 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
2123 // returns true if some parameters are negative
2124 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
2126 // expand parameter list
2128 for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
2130 for (ex count=*it-1; count > 0; count--) {
2134 } else if (*it < -1) {
2135 for (ex count=*it+1; count < 0; count++) {
2146 bool has_negative_parameters = false;
2148 for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
2154 m.append((*it+acc-1) * signum);
2156 m.append((*it-acc+1) * signum);
2162 has_negative_parameters = true;
2165 if (has_negative_parameters) {
2166 for (int i=0; i<m.nops(); i++) {
2168 m.let_op(i) = -m.op(i);
2176 return has_negative_parameters;
2180 // recursivly transforms H to corresponding multiple polylogarithms
2181 struct map_trafo_H_convert_to_Li : public map_function
2183 ex operator()(const ex& e)
2185 if (is_a<add>(e) || is_a<mul>(e)) {
2186 return e.map(*this);
2188 if (is_a<function>(e)) {
2189 std::string name = ex_to<function>(e).get_name();
2192 if (is_a<lst>(e.op(0))) {
2193 parameter = ex_to<lst>(e.op(0));
2195 parameter = lst(e.op(0));
2202 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2203 s.let_op(0) = s.op(0) * arg;
2204 return pf * Li(m, s).hold();
2206 for (int i=0; i<m.nops(); i++) {
2209 s.let_op(0) = s.op(0) * arg;
2210 return Li(m, s).hold();
2219 // recursivly transforms H to corresponding zetas
2220 struct map_trafo_H_convert_to_zeta : public map_function
2222 ex operator()(const ex& e)
2224 if (is_a<add>(e) || is_a<mul>(e)) {
2225 return e.map(*this);
2227 if (is_a<function>(e)) {
2228 std::string name = ex_to<function>(e).get_name();
2231 if (is_a<lst>(e.op(0))) {
2232 parameter = ex_to<lst>(e.op(0));
2234 parameter = lst(e.op(0));
2240 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2241 return pf * zeta(m, s);
2252 // remove trailing zeros from H-parameters
2253 struct map_trafo_H_reduce_trailing_zeros : public map_function
2255 ex operator()(const ex& e)
2257 if (is_a<add>(e) || is_a<mul>(e)) {
2258 return e.map(*this);
2260 if (is_a<function>(e)) {
2261 std::string name = ex_to<function>(e).get_name();
2264 if (is_a<lst>(e.op(0))) {
2265 parameter = ex_to<lst>(e.op(0));
2267 parameter = lst(e.op(0));
2270 if (parameter.op(parameter.nops()-1) == 0) {
2273 if (parameter.nops() == 1) {
2278 lst::const_iterator it = parameter.begin();
2279 while ((it != parameter.end()) && (*it == 0)) {
2282 if (it == parameter.end()) {
2283 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
2287 parameter.remove_last();
2288 int lastentry = parameter.nops();
2289 while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
2294 ex result = log(arg) * H(parameter,arg).hold();
2296 for (ex i=0; i<lastentry; i++) {
2297 if (parameter[i] > 0) {
2299 result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
2302 } else if (parameter[i] < 0) {
2304 result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
2312 if (lastentry < parameter.nops()) {
2313 result = result / (parameter.nops()-lastentry+1);
2314 return result.map(*this);
2326 // returns an expression with zeta functions corresponding to the parameter list for H
2327 ex convert_H_to_zeta(const lst& m)
2329 symbol xtemp("xtemp");
2330 map_trafo_H_reduce_trailing_zeros filter;
2331 map_trafo_H_convert_to_zeta filter2;
2332 return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
2336 // convert signs form Li to H representation
2337 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
2340 lst::const_iterator itm = m.begin();
2341 lst::const_iterator itx = ++x.begin();
2346 while (itx != x.end()) {
2347 signum *= (*itx > 0) ? 1 : -1;
2349 res.append((*itm) * signum);
2357 // multiplies an one-dimensional H with another H
2359 ex trafo_H_mult(const ex& h1, const ex& h2)
2364 ex h1nops = h1.op(0).nops();
2365 ex h2nops = h2.op(0).nops();
2367 hshort = h2.op(0).op(0);
2368 hlong = ex_to<lst>(h1.op(0));
2370 hshort = h1.op(0).op(0);
2372 hlong = ex_to<lst>(h2.op(0));
2374 hlong = h2.op(0).op(0);
2377 for (int i=0; i<=hlong.nops(); i++) {
2381 newparameter.append(hlong[j]);
2383 newparameter.append(hshort);
2384 for (; j<hlong.nops(); j++) {
2385 newparameter.append(hlong[j]);
2387 res += H(newparameter, h1.op(1)).hold();
2393 // applies trafo_H_mult recursively on expressions
2394 struct map_trafo_H_mult : public map_function
2396 ex operator()(const ex& e)
2399 return e.map(*this);
2407 for (int pos=0; pos<e.nops(); pos++) {
2408 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
2409 std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
2411 for (ex i=0; i<e.op(pos).op(1); i++) {
2412 Hlst.append(e.op(pos).op(0));
2416 } else if (is_a<function>(e.op(pos))) {
2417 std::string name = ex_to<function>(e.op(pos)).get_name();
2419 if (e.op(pos).op(0).nops() > 1) {
2422 Hlst.append(e.op(pos));
2427 result *= e.op(pos);
2430 if (Hlst.nops() > 0) {
2431 firstH = Hlst[Hlst.nops()-1];
2438 if (Hlst.nops() > 0) {
2439 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
2441 for (int i=1; i<Hlst.nops(); i++) {
2442 result *= Hlst.op(i);
2444 result = result.expand();
2445 map_trafo_H_mult recursion;
2446 return recursion(result);
2457 // do integration [ReV] (55)
2458 // put parameter 0 in front of existing parameters
2459 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
2463 if (is_a<function>(e)) {
2464 name = ex_to<function>(e).get_name();
2469 for (int i=0; i<e.nops(); i++) {
2470 if (is_a<function>(e.op(i))) {
2471 std::string name = ex_to<function>(e.op(i)).get_name();
2479 lst newparameter = ex_to<lst>(h.op(0));
2480 newparameter.prepend(0);
2481 ex addzeta = convert_H_to_zeta(newparameter);
2482 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2484 return e * (-H(lst(0),1/arg).hold());
2489 // do integration [ReV] (49)
2490 // put parameter 1 in front of existing parameters
2491 ex trafo_H_prepend_one(const ex& e, const ex& arg)
2495 if (is_a<function>(e)) {
2496 name = ex_to<function>(e).get_name();
2501 for (int i=0; i<e.nops(); i++) {
2502 if (is_a<function>(e.op(i))) {
2503 std::string name = ex_to<function>(e.op(i)).get_name();
2511 lst newparameter = ex_to<lst>(h.op(0));
2512 newparameter.prepend(1);
2513 return e.subs(h == H(newparameter, h.op(1)).hold());
2515 return e * H(lst(1),1-arg).hold();
2520 // do integration [ReV] (55)
2521 // put parameter -1 in front of existing parameters
2522 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
2526 if (is_a<function>(e)) {
2527 name = ex_to<function>(e).get_name();
2532 for (int i=0; i<e.nops(); i++) {
2533 if (is_a<function>(e.op(i))) {
2534 std::string name = ex_to<function>(e.op(i)).get_name();
2542 lst newparameter = ex_to<lst>(h.op(0));
2543 newparameter.prepend(-1);
2544 ex addzeta = convert_H_to_zeta(newparameter);
2545 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2547 ex addzeta = convert_H_to_zeta(lst(-1));
2548 return (e * (addzeta - H(lst(-1),1/arg).hold())).expand();
2553 // do integration [ReV] (55)
2554 // put parameter -1 in front of existing parameters
2555 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
2559 if (is_a<function>(e)) {
2560 name = ex_to<function>(e).get_name();
2565 for (int i=0; i<e.nops(); i++) {
2566 if (is_a<function>(e.op(i))) {
2567 std::string name = ex_to<function>(e.op(i)).get_name();
2575 lst newparameter = ex_to<lst>(h.op(0));
2576 newparameter.prepend(-1);
2577 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2579 return (e * H(lst(-1),(1-arg)/(1+arg)).hold()).expand();
2584 // do integration [ReV] (55)
2585 // put parameter 1 in front of existing parameters
2586 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
2590 if (is_a<function>(e)) {
2591 name = ex_to<function>(e).get_name();
2596 for (int i=0; i<e.nops(); i++) {
2597 if (is_a<function>(e.op(i))) {
2598 std::string name = ex_to<function>(e.op(i)).get_name();
2606 lst newparameter = ex_to<lst>(h.op(0));
2607 newparameter.prepend(1);
2608 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2610 return (e * H(lst(1),(1-arg)/(1+arg)).hold()).expand();
2615 // do x -> 1-x transformation
2616 struct map_trafo_H_1mx : public map_function
2618 ex operator()(const ex& e)
2620 if (is_a<add>(e) || is_a<mul>(e)) {
2621 return e.map(*this);
2624 if (is_a<function>(e)) {
2625 std::string name = ex_to<function>(e).get_name();
2628 lst parameter = ex_to<lst>(e.op(0));
2631 // special cases if all parameters are either 0, 1 or -1
2632 bool allthesame = true;
2633 if (parameter.op(0) == 0) {
2634 for (int i=1; i<parameter.nops(); i++) {
2635 if (parameter.op(i) != 0) {
2642 for (int i=parameter.nops(); i>0; i--) {
2643 newparameter.append(1);
2645 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2647 } else if (parameter.op(0) == -1) {
2648 throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
2650 for (int i=1; i<parameter.nops(); i++) {
2651 if (parameter.op(i) != 1) {
2658 for (int i=parameter.nops(); i>0; i--) {
2659 newparameter.append(0);
2661 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2665 lst newparameter = parameter;
2666 newparameter.remove_first();
2668 if (parameter.op(0) == 0) {
2671 ex res = convert_H_to_zeta(parameter);
2672 //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
2673 map_trafo_H_1mx recursion;
2674 ex buffer = recursion(H(newparameter, arg).hold());
2675 if (is_a<add>(buffer)) {
2676 for (int i=0; i<buffer.nops(); i++) {
2677 res -= trafo_H_prepend_one(buffer.op(i), arg);
2680 res -= trafo_H_prepend_one(buffer, arg);
2687 map_trafo_H_1mx recursion;
2688 map_trafo_H_mult unify;
2689 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
2691 while (parameter.op(firstzero) == 1) {
2694 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
2698 newparameter.append(parameter[j+1]);
2700 newparameter.append(1);
2701 for (; j<parameter.nops()-1; j++) {
2702 newparameter.append(parameter[j+1]);
2704 res -= H(newparameter, arg).hold();
2706 res = recursion(res).expand() / firstzero;
2716 // do x -> 1/x transformation
2717 struct map_trafo_H_1overx : public map_function
2719 ex operator()(const ex& e)
2721 if (is_a<add>(e) || is_a<mul>(e)) {
2722 return e.map(*this);
2725 if (is_a<function>(e)) {
2726 std::string name = ex_to<function>(e).get_name();
2729 lst parameter = ex_to<lst>(e.op(0));
2732 // special cases if all parameters are either 0, 1 or -1
2733 bool allthesame = true;
2734 if (parameter.op(0) == 0) {
2735 for (int i=1; i<parameter.nops(); i++) {
2736 if (parameter.op(i) != 0) {
2742 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
2744 } else if (parameter.op(0) == -1) {
2745 for (int i=1; i<parameter.nops(); i++) {
2746 if (parameter.op(i) != -1) {
2752 map_trafo_H_mult unify;
2753 return unify((pow(H(lst(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops())
2754 / factorial(parameter.nops())).expand());
2757 for (int i=1; i<parameter.nops(); i++) {
2758 if (parameter.op(i) != 1) {
2764 map_trafo_H_mult unify;
2765 return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() + H_polesign, parameter.nops())
2766 / factorial(parameter.nops())).expand());
2770 lst newparameter = parameter;
2771 newparameter.remove_first();
2773 if (parameter.op(0) == 0) {
2776 ex res = convert_H_to_zeta(parameter);
2777 map_trafo_H_1overx recursion;
2778 ex buffer = recursion(H(newparameter, arg).hold());
2779 if (is_a<add>(buffer)) {
2780 for (int i=0; i<buffer.nops(); i++) {
2781 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
2784 res += trafo_H_1tx_prepend_zero(buffer, arg);
2788 } else if (parameter.op(0) == -1) {
2790 // leading negative one
2791 ex res = convert_H_to_zeta(parameter);
2792 map_trafo_H_1overx recursion;
2793 ex buffer = recursion(H(newparameter, arg).hold());
2794 if (is_a<add>(buffer)) {
2795 for (int i=0; i<buffer.nops(); i++) {
2796 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
2799 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
2806 map_trafo_H_1overx recursion;
2807 map_trafo_H_mult unify;
2808 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
2810 while (parameter.op(firstzero) == 1) {
2813 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
2817 newparameter.append(parameter[j+1]);
2819 newparameter.append(1);
2820 for (; j<parameter.nops()-1; j++) {
2821 newparameter.append(parameter[j+1]);
2823 res -= H(newparameter, arg).hold();
2825 res = recursion(res).expand() / firstzero;
2837 // do x -> (1-x)/(1+x) transformation
2838 struct map_trafo_H_1mxt1px : public map_function
2840 ex operator()(const ex& e)
2842 if (is_a<add>(e) || is_a<mul>(e)) {
2843 return e.map(*this);
2846 if (is_a<function>(e)) {
2847 std::string name = ex_to<function>(e).get_name();
2850 lst parameter = ex_to<lst>(e.op(0));
2853 // special cases if all parameters are either 0, 1 or -1
2854 bool allthesame = true;
2855 if (parameter.op(0) == 0) {
2856 for (int i=1; i<parameter.nops(); i++) {
2857 if (parameter.op(i) != 0) {
2863 map_trafo_H_mult unify;
2864 return unify((pow(-H(lst(1),(1-arg)/(1+arg)).hold() - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
2865 / factorial(parameter.nops())).expand());
2867 } else if (parameter.op(0) == -1) {
2868 for (int i=1; i<parameter.nops(); i++) {
2869 if (parameter.op(i) != -1) {
2875 map_trafo_H_mult unify;
2876 return unify((pow(log(2) - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
2877 / factorial(parameter.nops())).expand());
2880 for (int i=1; i<parameter.nops(); i++) {
2881 if (parameter.op(i) != 1) {
2887 map_trafo_H_mult unify;
2888 return unify((pow(-log(2) - H(lst(0),(1-arg)/(1+arg)).hold() + H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
2889 / factorial(parameter.nops())).expand());
2893 lst newparameter = parameter;
2894 newparameter.remove_first();
2896 if (parameter.op(0) == 0) {
2899 ex res = convert_H_to_zeta(parameter);
2900 map_trafo_H_1mxt1px recursion;
2901 ex buffer = recursion(H(newparameter, arg).hold());
2902 if (is_a<add>(buffer)) {
2903 for (int i=0; i<buffer.nops(); i++) {
2904 res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
2907 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
2911 } else if (parameter.op(0) == -1) {
2913 // leading negative one
2914 ex res = convert_H_to_zeta(parameter);
2915 map_trafo_H_1mxt1px recursion;
2916 ex buffer = recursion(H(newparameter, arg).hold());
2917 if (is_a<add>(buffer)) {
2918 for (int i=0; i<buffer.nops(); i++) {
2919 res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
2922 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
2929 map_trafo_H_1mxt1px recursion;
2930 map_trafo_H_mult unify;
2931 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
2933 while (parameter.op(firstzero) == 1) {
2936 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
2940 newparameter.append(parameter[j+1]);
2942 newparameter.append(1);
2943 for (; j<parameter.nops()-1; j++) {
2944 newparameter.append(parameter[j+1]);
2946 res -= H(newparameter, arg).hold();
2948 res = recursion(res).expand() / firstzero;
2960 // do the actual summation.
2961 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
2963 const int j = m.size();
2965 std::vector<cln::cl_N> t(j);
2967 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2968 cln::cl_N factor = cln::expt(x, j) * one;
2974 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
2975 for (int k=j-2; k>=1; k--) {
2976 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
2978 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
2979 factor = factor * x;
2980 } while (t[0] != t0buf);
2986 } // end of anonymous namespace
2989 //////////////////////////////////////////////////////////////////////
2991 // Harmonic polylogarithm H(m,x)
2995 //////////////////////////////////////////////////////////////////////
2998 static ex H_evalf(const ex& x1, const ex& x2)
3000 if (is_a<lst>(x1)) {
3003 if (is_a<numeric>(x2)) {
3004 x = ex_to<numeric>(x2).to_cl_N();
3006 ex x2_val = x2.evalf();
3007 if (is_a<numeric>(x2_val)) {
3008 x = ex_to<numeric>(x2_val).to_cl_N();
3012 for (int i=0; i<x1.nops(); i++) {
3013 if (!x1.op(i).info(info_flags::integer)) {
3014 return H(x1, x2).hold();
3017 if (x1.nops() < 1) {
3018 return H(x1, x2).hold();
3021 const lst& morg = ex_to<lst>(x1);
3022 // remove trailing zeros ...
3023 if (*(--morg.end()) == 0) {
3024 symbol xtemp("xtemp");
3025 map_trafo_H_reduce_trailing_zeros filter;
3026 return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
3028 // ... and expand parameter notation
3029 bool has_minus_one = false;
3031 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
3033 for (ex count=*it-1; count > 0; count--) {
3037 } else if (*it <= -1) {
3038 for (ex count=*it+1; count < 0; count++) {
3042 has_minus_one = true;
3049 if (cln::abs(x) < 0.95) {
3053 if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
3054 // negative parameters -> s_lst is filled
3055 std::vector<int> m_int;
3056 std::vector<cln::cl_N> x_cln;
3057 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
3058 it_int != m_lst.end(); it_int++, it_cln++) {
3059 m_int.push_back(ex_to<numeric>(*it_int).to_int());
3060 x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
3062 x_cln.front() = x_cln.front() * x;
3063 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
3065 // only positive parameters
3067 if (m_lst.nops() == 1) {
3068 return Li(m_lst.op(0), x2).evalf();
3070 std::vector<int> m_int;
3071 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
3072 m_int.push_back(ex_to<numeric>(*it).to_int());
3074 return numeric(H_do_sum(m_int, x));
3078 symbol xtemp("xtemp");
3081 // ensure that the realpart of the argument is positive
3082 if (cln::realpart(x) < 0) {
3084 for (int i=0; i<m.nops(); i++) {
3086 m.let_op(i) = -m.op(i);
3093 if (cln::abs(x) >= 2.0) {
3094 map_trafo_H_1overx trafo;
3095 res *= trafo(H(m, xtemp));
3096 if (cln::imagpart(x) <= 0) {
3097 res = res.subs(H_polesign == -I*Pi);
3099 res = res.subs(H_polesign == I*Pi);
3101 return res.subs(xtemp == numeric(x)).evalf();
3104 // check transformations for 0.95 <= |x| < 2.0
3106 // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47
3107 if (cln::abs(x-9.53) <= 9.47) {
3109 map_trafo_H_1mxt1px trafo;
3110 res *= trafo(H(m, xtemp));
3113 if (has_minus_one) {
3114 map_trafo_H_convert_to_Li filter;
3115 return filter(H(m, numeric(x)).hold()).evalf();
3117 map_trafo_H_1mx trafo;
3118 res *= trafo(H(m, xtemp));
3121 return res.subs(xtemp == numeric(x)).evalf();
3124 return H(x1,x2).hold();
3128 static ex H_eval(const ex& m_, const ex& x)
3131 if (is_a<lst>(m_)) {
3136 if (m.nops() == 0) {
3144 if (*m.begin() > _ex1) {
3150 } else if (*m.begin() < _ex_1) {
3156 } else if (*m.begin() == _ex0) {
3163 for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
3164 if ((*it).info(info_flags::integer)) {
3175 } else if (*it < _ex_1) {
3195 } else if (step == 1) {
3207 // if some m_i is not an integer
3208 return H(m_, x).hold();
3211 if ((x == _ex1) && (*(--m.end()) != _ex0)) {
3212 return convert_H_to_zeta(m);
3218 return H(m_, x).hold();
3220 return pow(log(x), m.nops()) / factorial(m.nops());
3223 return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
3225 } else if ((step == 1) && (pos1 == _ex0)){
3230 return pow(-1, p) * S(n, p, -x);
3236 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
3237 return H(m_, x).evalf();
3239 return H(m_, x).hold();
3243 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
3246 seq.push_back(expair(H(m, x), 0));
3247 return pseries(rel, seq);
3251 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
3253 GINAC_ASSERT(deriv_param < 2);
3254 if (deriv_param == 0) {
3258 if (is_a<lst>(m_)) {
3274 return 1/(1-x) * H(m, x);
3275 } else if (mb == _ex_1) {
3276 return 1/(1+x) * H(m, x);
3283 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
3286 if (is_a<lst>(m_)) {
3291 c.s << "\\mbox{H}_{";
3292 lst::const_iterator itm = m.begin();
3295 for (; itm != m.end(); itm++) {
3305 REGISTER_FUNCTION(H,
3306 evalf_func(H_evalf).
3308 series_func(H_series).
3309 derivative_func(H_deriv).
3310 print_func<print_latex>(H_print_latex).
3311 do_not_evalf_params());
3314 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
3315 ex convert_H_to_Li(const ex& m, const ex& x)
3317 map_trafo_H_reduce_trailing_zeros filter;
3318 map_trafo_H_convert_to_Li filter2;
3320 return filter2(filter(H(m, x).hold()));
3322 return filter2(filter(H(lst(m), x).hold()));
3327 //////////////////////////////////////////////////////////////////////
3329 // Multiple zeta values zeta(x) and zeta(x,s)
3333 //////////////////////////////////////////////////////////////////////
3336 // anonymous namespace for helper functions
3340 // parameters and data for [Cra] algorithm
3341 const cln::cl_N lambda = cln::cl_N("319/320");
3344 std::vector<std::vector<cln::cl_N> > f_kj;
3345 std::vector<cln::cl_N> crB;
3346 std::vector<std::vector<cln::cl_N> > crG;
3347 std::vector<cln::cl_N> crX;
3350 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
3352 const int size = a.size();
3353 for (int n=0; n<size; n++) {
3355 for (int m=0; m<=n; m++) {
3356 c[n] = c[n] + a[m]*b[n-m];
3363 void initcX(const std::vector<int>& s)
3365 const int k = s.size();
3371 for (int i=0; i<=L2; i++) {
3372 crB.push_back(bernoulli(i).to_cl_N() / cln::factorial(i));
3377 for (int m=0; m<k-1; m++) {
3378 std::vector<cln::cl_N> crGbuf;
3381 for (int i=0; i<=L2; i++) {
3382 crGbuf.push_back(cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2));
3384 crG.push_back(crGbuf);
3389 for (int m=0; m<k-1; m++) {
3390 std::vector<cln::cl_N> Xbuf;
3391 for (int i=0; i<=L2; i++) {
3392 Xbuf.push_back(crX[i] * crG[m][i]);
3394 halfcyclic_convolute(Xbuf, crB, crX);
3400 cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk)
3402 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3403 cln::cl_N factor = cln::expt(lambda, Sqk);
3404 cln::cl_N res = factor / Sqk * crX[0] * one;
3409 factor = factor * lambda;
3411 res = res + crX[N] * factor / (N+Sqk);
3412 } while ((res != resbuf) || cln::zerop(crX[N]));
3418 void calc_f(int maxr)
3423 cln::cl_N t0, t1, t2, t3, t4;
3425 std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
3426 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3428 t0 = cln::exp(-lambda);
3430 for (k=1; k<=L1; k++) {
3433 for (j=1; j<=maxr; j++) {
3436 for (i=2; i<=j; i++) {
3440 (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
3448 cln::cl_N crandall_Z(const std::vector<int>& s)
3450 const int j = s.size();
3459 t0 = t0 + f_kj[q+j-2][s[0]-1];
3460 } while (t0 != t0buf);
3462 return t0 / cln::factorial(s[0]-1);
3465 std::vector<cln::cl_N> t(j);
3472 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
3473 for (int k=j-2; k>=1; k--) {
3474 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
3476 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
3477 } while (t[0] != t0buf);
3479 return t[0] / cln::factorial(s[0]-1);
3484 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
3486 std::vector<int> r = s;
3487 const int j = r.size();
3489 // decide on maximal size of f_kj for crandall_Z
3493 L1 = Digits * 3 + j*2;
3496 // decide on maximal size of crX for crandall_Y
3499 } else if (Digits < 86) {
3501 } else if (Digits < 192) {
3503 } else if (Digits < 394) {
3505 } else if (Digits < 808) {
3515 for (int i=0; i<j; i++) {
3524 const cln::cl_N r0factorial = cln::factorial(r[0]-1);
3526 std::vector<int> rz;
3529 for (int k=r.size()-1; k>0; k--) {
3531 rz.insert(rz.begin(), r.back());
3532 skp1buf = rz.front();
3538 for (int q=0; q<skp1buf; q++) {
3540 cln::cl_N pp1 = crandall_Y_loop(Srun+q-k);
3541 cln::cl_N pp2 = crandall_Z(rz);
3546 res = res - pp1 * pp2 / cln::factorial(q);
3548 res = res + pp1 * pp2 / cln::factorial(q);
3551 rz.front() = skp1buf;
3553 rz.insert(rz.begin(), r.back());
3557 res = (res + crandall_Y_loop(S-j)) / r0factorial + crandall_Z(rz);
3563 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
3565 const int j = r.size();
3567 // buffer for subsums
3568 std::vector<cln::cl_N> t(j);
3569 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3576 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
3577 for (int k=j-2; k>=0; k--) {
3578 t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
3580 } while (t[0] != t0buf);
3586 // does Hoelder convolution. see [BBB] (7.0)
3587 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
3589 // prepare parameters
3590 // holds Li arguments in [BBB] notation
3591 std::vector<int> s = s_;
3592 std::vector<int> m_p = m_;
3593 std::vector<int> m_q;
3594 // holds Li arguments in nested sums notation
3595 std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
3596 s_p[0] = s_p[0] * cln::cl_N("1/2");
3597 // convert notations
3599 for (int i=0; i<s_.size(); i++) {
3604 s[i] = sig * std::abs(s[i]);
3606 std::vector<cln::cl_N> s_q;
3607 cln::cl_N signum = 1;
3610 cln::cl_N res = multipleLi_do_sum(m_p, s_p);
3615 // change parameters
3616 if (s.front() > 0) {
3617 if (m_p.front() == 1) {
3618 m_p.erase(m_p.begin());
3619 s_p.erase(s_p.begin());
3620 if (s_p.size() > 0) {
3621 s_p.front() = s_p.front() * cln::cl_N("1/2");
3627 m_q.insert(m_q.begin(), 1);
3628 if (s_q.size() > 0) {
3629 s_q.front() = s_q.front() * 2;
3631 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3634 if (m_p.front() == 1) {
3635 m_p.erase(m_p.begin());
3636 cln::cl_N spbuf = s_p.front();
3637 s_p.erase(s_p.begin());
3638 if (s_p.size() > 0) {
3639 s_p.front() = s_p.front() * spbuf;
3642 m_q.insert(m_q.begin(), 1);
3643 if (s_q.size() > 0) {
3644 s_q.front() = s_q.front() * 4;
3646 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
3650 m_q.insert(m_q.begin(), 1);
3651 if (s_q.size() > 0) {
3652 s_q.front() = s_q.front() * 2;
3654 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3659 if (m_p.size() == 0) break;
3661 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
3666 res = res + signum * multipleLi_do_sum(m_q, s_q);
3672 } // end of anonymous namespace
3675 //////////////////////////////////////////////////////////////////////
3677 // Multiple zeta values zeta(x)
3681 //////////////////////////////////////////////////////////////////////
3684 static ex zeta1_evalf(const ex& x)
3686 if (is_exactly_a<lst>(x) && (x.nops()>1)) {
3688 // multiple zeta value
3689 const int count = x.nops();
3690 const lst& xlst = ex_to<lst>(x);
3691 std::vector<int> r(count);
3693 // check parameters and convert them
3694 lst::const_iterator it1 = xlst.begin();
3695 std::vector<int>::iterator it2 = r.begin();
3697 if (!(*it1).info(info_flags::posint)) {
3698 return zeta(x).hold();
3700 *it2 = ex_to<numeric>(*it1).to_int();
3703 } while (it2 != r.end());
3705 // check for divergence
3707 return zeta(x).hold();
3710 // decide on summation algorithm
3711 // this is still a bit clumsy
3712 int limit = (Digits>17) ? 10 : 6;
3713 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
3714 return numeric(zeta_do_sum_Crandall(r));
3716 return numeric(zeta_do_sum_simple(r));
3720 // single zeta value
3721 if (is_exactly_a<numeric>(x) && (x != 1)) {
3723 return zeta(ex_to<numeric>(x));
3724 } catch (const dunno &e) { }
3727 return zeta(x).hold();
3731 static ex zeta1_eval(const ex& m)
3733 if (is_exactly_a<lst>(m)) {
3734 if (m.nops() == 1) {
3735 return zeta(m.op(0));
3737 return zeta(m).hold();
3740 if (m.info(info_flags::numeric)) {
3741 const numeric& y = ex_to<numeric>(m);
3742 // trap integer arguments:
3743 if (y.is_integer()) {
3747 if (y.is_equal(*_num1_p)) {
3748 return zeta(m).hold();
3750 if (y.info(info_flags::posint)) {
3751 if (y.info(info_flags::odd)) {
3752 return zeta(m).hold();
3754 return abs(bernoulli(y)) * pow(Pi, y) * pow(*_num2_p, y-(*_num1_p)) / factorial(y);
3757 if (y.info(info_flags::odd)) {
3758 return -bernoulli((*_num1_p)-y) / ((*_num1_p)-y);
3765 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
3766 return zeta1_evalf(m);
3769 return zeta(m).hold();
3773 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
3775 GINAC_ASSERT(deriv_param==0);
3777 if (is_exactly_a<lst>(m)) {
3780 return zetaderiv(_ex1, m);
3785 static void zeta1_print_latex(const ex& m_, const print_context& c)
3788 if (is_a<lst>(m_)) {
3789 const lst& m = ex_to<lst>(m_);
3790 lst::const_iterator it = m.begin();
3793 for (; it != m.end(); it++) {
3804 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1).
3805 evalf_func(zeta1_evalf).
3806 eval_func(zeta1_eval).
3807 derivative_func(zeta1_deriv).
3808 print_func<print_latex>(zeta1_print_latex).
3809 do_not_evalf_params().
3813 //////////////////////////////////////////////////////////////////////
3815 // Alternating Euler sum zeta(x,s)
3819 //////////////////////////////////////////////////////////////////////
3822 static ex zeta2_evalf(const ex& x, const ex& s)
3824 if (is_exactly_a<lst>(x)) {
3826 // alternating Euler sum
3827 const int count = x.nops();
3828 const lst& xlst = ex_to<lst>(x);
3829 const lst& slst = ex_to<lst>(s);
3830 std::vector<int> xi(count);
3831 std::vector<int> si(count);
3833 // check parameters and convert them
3834 lst::const_iterator it_xread = xlst.begin();
3835 lst::const_iterator it_sread = slst.begin();
3836 std::vector<int>::iterator it_xwrite = xi.begin();
3837 std::vector<int>::iterator it_swrite = si.begin();
3839 if (!(*it_xread).info(info_flags::posint)) {
3840 return zeta(x, s).hold();
3842 *it_xwrite = ex_to<numeric>(*it_xread).to_int();
3843 if (*it_sread > 0) {
3852 } while (it_xwrite != xi.end());
3854 // check for divergence
3855 if ((xi[0] == 1) && (si[0] == 1)) {
3856 return zeta(x, s).hold();
3859 // use Hoelder convolution
3860 return numeric(zeta_do_Hoelder_convolution(xi, si));
3863 return zeta(x, s).hold();
3867 static ex zeta2_eval(const ex& m, const ex& s_)
3869 if (is_exactly_a<lst>(s_)) {
3870 const lst& s = ex_to<lst>(s_);
3871 for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
3872 if ((*it).info(info_flags::positive)) {
3875 return zeta(m, s_).hold();
3878 } else if (s_.info(info_flags::positive)) {
3882 return zeta(m, s_).hold();
3886 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
3888 GINAC_ASSERT(deriv_param==0);
3890 if (is_exactly_a<lst>(m)) {
3893 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
3894 return zetaderiv(_ex1, m);
3901 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
3904 if (is_a<lst>(m_)) {
3910 if (is_a<lst>(s_)) {
3916 lst::const_iterator itm = m.begin();
3917 lst::const_iterator its = s.begin();
3919 c.s << "\\overline{";
3927 for (; itm != m.end(); itm++, its++) {
3930 c.s << "\\overline{";
3941 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2).
3942 evalf_func(zeta2_evalf).
3943 eval_func(zeta2_eval).
3944 derivative_func(zeta2_deriv).
3945 print_func<print_latex>(zeta2_print_latex).
3946 do_not_evalf_params().
3950 } // namespace GiNaC