1 /** @file inifcns_gamma.cpp
3 * Implementation of Gamma-function, Beta-function, Polygamma-functions, and
4 * some related stuff. */
7 * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
9 * This program is free software; you can redistribute it and/or modify
10 * it under the terms of the GNU General Public License as published by
11 * the Free Software Foundation; either version 2 of the License, or
12 * (at your option) any later version.
14 * This program is distributed in the hope that it will be useful,
15 * but WITHOUT ANY WARRANTY; without even the implied warranty of
16 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
17 * GNU General Public License for more details.
19 * You should have received a copy of the GNU General Public License
20 * along with this program; if not, write to the Free Software
21 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
32 #include "relational.h"
40 // Logarithm of Gamma function
43 static ex lgamma_evalf(const ex & x)
47 END_TYPECHECK(lgamma(x))
49 return lgamma(ex_to<numeric>(x));
53 /** Evaluation of lgamma(x), the natural logarithm of the Gamma function.
54 * Knows about integer arguments and that's it. Somebody ought to provide
55 * some good numerical evaluation some day...
57 * @exception GiNaC::pole_error("lgamma_eval(): logarithmic pole",0) */
58 static ex lgamma_eval(const ex & x)
60 if (x.info(info_flags::numeric)) {
61 // trap integer arguments:
62 if (x.info(info_flags::integer)) {
63 // lgamma(n) -> log((n-1)!) for postitive n
64 if (x.info(info_flags::posint))
65 return log(factorial(x + _ex_1()));
67 throw (pole_error("lgamma_eval(): logarithmic pole",0));
69 // lgamma_evalf should be called here once it becomes available
72 return lgamma(x).hold();
76 static ex lgamma_deriv(const ex & x, unsigned deriv_param)
78 GINAC_ASSERT(deriv_param==0);
80 // d/dx lgamma(x) -> psi(x)
85 static ex lgamma_series(const ex & arg,
86 const relational & rel,
91 // Taylor series where there is no pole falls back to psi function
93 // On a pole at -m we could use the recurrence relation
94 // lgamma(x) == lgamma(x+1)-log(x)
96 // series(lgamma(x),x==-m,order) ==
97 // series(lgamma(x+m+1)-log(x)...-log(x+m)),x==-m,order);
98 const ex arg_pt = arg.subs(rel);
99 if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
100 throw do_taylor(); // caught by function::series()
101 // if we got here we have to care for a simple pole of tgamma(-m):
102 numeric m = -ex_to<numeric>(arg_pt);
104 for (numeric p = 0; p<=m; ++p)
106 return (lgamma(arg+m+_ex1())-recur).series(rel, order, options);
110 REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval).
111 evalf_func(lgamma_evalf).
112 derivative_func(lgamma_deriv).
113 series_func(lgamma_series).
114 latex_name("\\log \\Gamma"));
118 // true Gamma function
121 static ex tgamma_evalf(const ex & x)
125 END_TYPECHECK(tgamma(x))
127 return tgamma(ex_to<numeric>(x));
131 /** Evaluation of tgamma(x), the true Gamma function. Knows about integer
132 * arguments, half-integer arguments and that's it. Somebody ought to provide
133 * some good numerical evaluation some day...
135 * @exception pole_error("tgamma_eval(): simple pole",0) */
136 static ex tgamma_eval(const ex & x)
138 if (x.info(info_flags::numeric)) {
139 // trap integer arguments:
140 const numeric two_x = _num2()*ex_to<numeric>(x);
141 if (two_x.is_even()) {
142 // tgamma(n) -> (n-1)! for postitive n
143 if (two_x.is_positive()) {
144 return factorial(ex_to<numeric>(x).sub(_num1()));
146 throw (pole_error("tgamma_eval(): simple pole",1));
149 // trap half integer arguments:
150 if (two_x.is_integer()) {
151 // trap positive x==(n+1/2)
152 // tgamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n)
153 if (two_x.is_positive()) {
154 const numeric n = ex_to<numeric>(x).sub(_num1_2());
155 return (doublefactorial(n.mul(_num2()).sub(_num1())).div(pow(_num2(),n))) * pow(Pi,_ex1_2());
157 // trap negative x==(-n+1/2)
158 // tgamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1))
159 const numeric n = abs(ex_to<numeric>(x).sub(_num1_2()));
160 return (pow(_num_2(), n).div(doublefactorial(n.mul(_num2()).sub(_num1()))))*power(Pi,_ex1_2());
163 // tgamma_evalf should be called here once it becomes available
166 return tgamma(x).hold();
170 static ex tgamma_deriv(const ex & x, unsigned deriv_param)
172 GINAC_ASSERT(deriv_param==0);
174 // d/dx tgamma(x) -> psi(x)*tgamma(x)
175 return psi(x)*tgamma(x);
179 static ex tgamma_series(const ex & arg,
180 const relational & rel,
185 // Taylor series where there is no pole falls back to psi function
187 // On a pole at -m use the recurrence relation
188 // tgamma(x) == tgamma(x+1) / x
189 // from which follows
190 // series(tgamma(x),x==-m,order) ==
191 // series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x==-m,order+1);
192 const ex arg_pt = arg.subs(rel);
193 if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
194 throw do_taylor(); // caught by function::series()
195 // if we got here we have to care for a simple pole at -m:
196 const numeric m = -ex_to<numeric>(arg_pt);
197 ex ser_denom = _ex1();
198 for (numeric p; p<=m; ++p)
200 return (tgamma(arg+m+_ex1())/ser_denom).series(rel, order+1, options);
204 REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval).
205 evalf_func(tgamma_evalf).
206 derivative_func(tgamma_deriv).
207 series_func(tgamma_series).
208 latex_name("\\Gamma"));
215 static ex beta_evalf(const ex & x, const ex & y)
220 END_TYPECHECK(beta(x,y))
222 return tgamma(ex_to<numeric>(x))*tgamma(ex_to<numeric>(y))/tgamma(ex_to<numeric>(x+y));
226 static ex beta_eval(const ex & x, const ex & y)
228 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
229 // treat all problematic x and y that may not be passed into tgamma,
230 // because they would throw there although beta(x,y) is well-defined
231 // using the formula beta(x,y) == (-1)^y * beta(1-x-y, y)
232 const numeric nx = ex_to<numeric>(x);
233 const numeric ny = ex_to<numeric>(y);
234 if (nx.is_real() && nx.is_integer() &&
235 ny.is_real() && ny.is_integer()) {
236 if (nx.is_negative()) {
238 return pow(_num_1(), ny)*beta(1-x-y, y);
240 throw (pole_error("beta_eval(): simple pole",1));
242 if (ny.is_negative()) {
244 return pow(_num_1(), nx)*beta(1-y-x, x);
246 throw (pole_error("beta_eval(): simple pole",1));
248 return tgamma(x)*tgamma(y)/tgamma(x+y);
250 // no problem in numerator, but denominator has pole:
251 if ((nx+ny).is_real() &&
252 (nx+ny).is_integer() &&
253 !(nx+ny).is_positive())
256 return tgamma(x)*tgamma(y)/tgamma(x+y);
259 return beta(x,y).hold();
263 static ex beta_deriv(const ex & x, const ex & y, unsigned deriv_param)
265 GINAC_ASSERT(deriv_param<2);
268 // d/dx beta(x,y) -> (psi(x)-psi(x+y)) * beta(x,y)
270 retval = (psi(x)-psi(x+y))*beta(x,y);
271 // d/dy beta(x,y) -> (psi(y)-psi(x+y)) * beta(x,y)
273 retval = (psi(y)-psi(x+y))*beta(x,y);
278 static ex beta_series(const ex & arg1,
280 const relational & rel,
285 // Taylor series where there is no pole of one of the tgamma functions
286 // falls back to beta function evaluation. Otherwise, fall back to
287 // tgamma series directly.
288 const ex arg1_pt = arg1.subs(rel);
289 const ex arg2_pt = arg2.subs(rel);
290 GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol));
291 const symbol *s = static_cast<symbol *>(rel.lhs().bp);
292 ex arg1_ser, arg2_ser, arg1arg2_ser;
293 if ((!arg1_pt.info(info_flags::integer) || arg1_pt.info(info_flags::positive)) &&
294 (!arg2_pt.info(info_flags::integer) || arg2_pt.info(info_flags::positive)))
295 throw do_taylor(); // caught by function::series()
296 // trap the case where arg1 is on a pole:
297 if (arg1.info(info_flags::integer) && !arg1.info(info_flags::positive))
298 arg1_ser = tgamma(arg1+*s).series(rel, order, options);
300 arg1_ser = tgamma(arg1).series(rel,order);
301 // trap the case where arg2 is on a pole:
302 if (arg2.info(info_flags::integer) && !arg2.info(info_flags::positive))
303 arg2_ser = tgamma(arg2+*s).series(rel, order, options);
305 arg2_ser = tgamma(arg2).series(rel,order);
306 // trap the case where arg1+arg2 is on a pole:
307 if ((arg1+arg2).info(info_flags::integer) && !(arg1+arg2).info(info_flags::positive))
308 arg1arg2_ser = tgamma(arg2+arg1+*s).series(rel, order, options);
310 arg1arg2_ser = tgamma(arg2+arg1).series(rel,order);
311 // compose the result (expanding all the terms):
312 return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand();
316 REGISTER_FUNCTION(beta, eval_func(beta_eval).
317 evalf_func(beta_evalf).
318 derivative_func(beta_deriv).
319 series_func(beta_series).
320 latex_name("\\mbox{B}").
321 set_symmetry(sy_symm(0, 1)));
325 // Psi-function (aka digamma-function)
328 static ex psi1_evalf(const ex & x)
332 END_TYPECHECK(psi(x))
334 return psi(ex_to<numeric>(x));
337 /** Evaluation of digamma-function psi(x).
338 * Somebody ought to provide some good numerical evaluation some day... */
339 static ex psi1_eval(const ex & x)
341 if (x.info(info_flags::numeric)) {
342 const numeric nx = ex_to<numeric>(x);
343 if (nx.is_integer()) {
345 if (nx.is_positive()) {
346 // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - Euler
348 for (numeric i(nx+_num_1()); i>0; --i)
352 // for non-positive integers there is a pole:
353 throw (pole_error("psi_eval(): simple pole",1));
356 if ((_num2()*nx).is_integer()) {
358 if (nx.is_positive()) {
359 // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - Euler - 2log(2)
361 for (numeric i = (nx+_num_1())*_num2(); i>0; i-=_num2())
362 rat += _num2()*i.inverse();
363 return rat-Euler-_ex2()*log(_ex2());
365 // use the recurrence relation
366 // psi(-m-1/2) == psi(-m-1/2+1) - 1 / (-m-1/2)
367 // to relate psi(-m-1/2) to psi(1/2):
368 // psi(-m-1/2) == psi(1/2) + r
369 // where r == ((-1/2)^(-1) + ... + (-m-1/2)^(-1))
371 for (numeric p = nx; p<0; ++p)
372 recur -= pow(p, _num_1());
373 return recur+psi(_ex1_2());
376 // psi1_evalf should be called here once it becomes available
379 return psi(x).hold();
382 static ex psi1_deriv(const ex & x, unsigned deriv_param)
384 GINAC_ASSERT(deriv_param==0);
386 // d/dx psi(x) -> psi(1,x)
387 return psi(_ex1(), x);
390 static ex psi1_series(const ex & arg,
391 const relational & rel,
396 // Taylor series where there is no pole falls back to polygamma function
398 // On a pole at -m use the recurrence relation
399 // psi(x) == psi(x+1) - 1/z
400 // from which follows
401 // series(psi(x),x==-m,order) ==
402 // series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x==-m,order);
403 const ex arg_pt = arg.subs(rel);
404 if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
405 throw do_taylor(); // caught by function::series()
406 // if we got here we have to care for a simple pole at -m:
407 const numeric m = -ex_to<numeric>(arg_pt);
409 for (numeric p; p<=m; ++p)
410 recur += power(arg+p,_ex_1());
411 return (psi(arg+m+_ex1())-recur).series(rel, order, options);
414 const unsigned function_index_psi1 =
415 function::register_new(function_options("psi").
416 eval_func(psi1_eval).
417 evalf_func(psi1_evalf).
418 derivative_func(psi1_deriv).
419 series_func(psi1_series).
424 // Psi-functions (aka polygamma-functions) psi(0,x)==psi(x)
427 static ex psi2_evalf(const ex & n, const ex & x)
432 END_TYPECHECK(psi(n,x))
434 return psi(ex_to<numeric>(n), ex_to<numeric>(x));
437 /** Evaluation of polygamma-function psi(n,x).
438 * Somebody ought to provide some good numerical evaluation some day... */
439 static ex psi2_eval(const ex & n, const ex & x)
441 // psi(0,x) -> psi(x)
444 // psi(-1,x) -> log(tgamma(x))
445 if (n.is_equal(_ex_1()))
446 return log(tgamma(x));
447 if (n.info(info_flags::numeric) && n.info(info_flags::posint) &&
448 x.info(info_flags::numeric)) {
449 const numeric nn = ex_to<numeric>(n);
450 const numeric nx = ex_to<numeric>(x);
451 if (nx.is_integer()) {
453 if (nx.is_equal(_num1()))
454 // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1)
455 return pow(_num_1(),nn+_num1())*factorial(nn)*zeta(ex(nn+_num1()));
456 if (nx.is_positive()) {
457 // use the recurrence relation
458 // psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1)
459 // to relate psi(n,m) to psi(n,1):
460 // psi(n,m) == psi(n,1) + r
461 // where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1))
463 for (numeric p = 1; p<nx; ++p)
464 recur += pow(p, -nn+_num_1());
465 recur *= factorial(nn)*pow(_num_1(), nn);
466 return recur+psi(n,_ex1());
468 // for non-positive integers there is a pole:
469 throw (pole_error("psi2_eval(): pole",1));
472 if ((_num2()*nx).is_integer()) {
474 if (nx.is_equal(_num1_2()))
475 // use psi(n,1/2) == (-)^(n+1) * n! * (2^(n+1)-1) * zeta(n+1)
476 return pow(_num_1(),nn+_num1())*factorial(nn)*(pow(_num2(),nn+_num1()) + _num_1())*zeta(ex(nn+_num1()));
477 if (nx.is_positive()) {
478 const numeric m = nx - _num1_2();
479 // use the multiplication formula
480 // psi(n,2*m) == (psi(n,m) + psi(n,m+1/2)) / 2^(n+1)
481 // to revert to positive integer case
482 return psi(n,_num2()*m)*pow(_num2(),nn+_num1())-psi(n,m);
484 // use the recurrence relation
485 // psi(n,-m-1/2) == psi(n,-m-1/2+1) - (-)^n * n! / (-m-1/2)^(n+1)
486 // to relate psi(n,-m-1/2) to psi(n,1/2):
487 // psi(n,-m-1/2) == psi(n,1/2) + r
488 // where r == (-)^(n+1) * n! * ((-1/2)^(-n-1) + ... + (-m-1/2)^(-n-1))
490 for (numeric p = nx; p<0; ++p)
491 recur += pow(p, -nn+_num_1());
492 recur *= factorial(nn)*pow(_num_1(), nn+_num_1());
493 return recur+psi(n,_ex1_2());
496 // psi2_evalf should be called here once it becomes available
499 return psi(n, x).hold();
502 static ex psi2_deriv(const ex & n, const ex & x, unsigned deriv_param)
504 GINAC_ASSERT(deriv_param<2);
506 if (deriv_param==0) {
508 throw(std::logic_error("cannot diff psi(n,x) with respect to n"));
510 // d/dx psi(n,x) -> psi(n+1,x)
511 return psi(n+_ex1(), x);
514 static ex psi2_series(const ex & n,
516 const relational & rel,
521 // Taylor series where there is no pole falls back to polygamma function
523 // On a pole at -m use the recurrence relation
524 // psi(n,x) == psi(n,x+1) - (-)^n * n! / x^(n+1)
525 // from which follows
526 // series(psi(x),x==-m,order) ==
527 // series(psi(x+m+1) - (-1)^n * n! * ((x)^(-n-1) + (x+1)^(-n-1) + ...
528 // ... + (x+m)^(-n-1))),x==-m,order);
529 const ex arg_pt = arg.subs(rel);
530 if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
531 throw do_taylor(); // caught by function::series()
532 // if we got here we have to care for a pole of order n+1 at -m:
533 const numeric m = -ex_to<numeric>(arg_pt);
535 for (numeric p; p<=m; ++p)
536 recur += power(arg+p,-n+_ex_1());
537 recur *= factorial(n)*power(_ex_1(),n);
538 return (psi(n, arg+m+_ex1())-recur).series(rel, order, options);
541 const unsigned function_index_psi2 =
542 function::register_new(function_options("psi").
543 eval_func(psi2_eval).
544 evalf_func(psi2_evalf).
545 derivative_func(psi2_deriv).
546 series_func(psi2_series).