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"
39 // Logarithm of Gamma function
42 static ex lgamma_evalf(const ex & x)
46 END_TYPECHECK(lgamma(x))
48 return lgamma(ex_to_numeric(x));
52 /** Evaluation of lgamma(x), the natural logarithm of the Gamma function.
53 * Knows about integer arguments and that's it. Somebody ought to provide
54 * some good numerical evaluation some day...
56 * @exception GiNaC::pole_error("lgamma_eval(): logarithmic pole",0) */
57 static ex lgamma_eval(const ex & x)
59 if (x.info(info_flags::numeric)) {
60 // trap integer arguments:
61 if (x.info(info_flags::integer)) {
62 // lgamma(n) -> log((n-1)!) for postitive n
63 if (x.info(info_flags::posint))
64 return log(factorial(x.exadd(_ex_1())));
66 throw (pole_error("lgamma_eval(): logarithmic pole",0));
68 // lgamma_evalf should be called here once it becomes available
71 return lgamma(x).hold();
75 static ex lgamma_deriv(const ex & x, unsigned deriv_param)
77 GINAC_ASSERT(deriv_param==0);
79 // d/dx lgamma(x) -> psi(x)
84 static ex lgamma_series(const ex & arg,
85 const relational & rel,
90 // Taylor series where there is no pole falls back to psi function
92 // On a pole at -m we could use the recurrence relation
93 // lgamma(x) == lgamma(x+1)-log(x)
95 // series(lgamma(x),x==-m,order) ==
96 // series(lgamma(x+m+1)-log(x)...-log(x+m)),x==-m,order);
97 const ex arg_pt = arg.subs(rel);
98 if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
99 throw do_taylor(); // caught by function::series()
100 // if we got here we have to care for a simple pole of tgamma(-m):
101 numeric m = -ex_to_numeric(arg_pt);
103 for (numeric p; p<=m; ++p)
105 return (lgamma(arg+m+_ex1())-recur).series(rel, order, options);
109 REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval).
110 evalf_func(lgamma_evalf).
111 derivative_func(lgamma_deriv).
112 series_func(lgamma_series));
116 // true Gamma function
119 static ex tgamma_evalf(const ex & x)
123 END_TYPECHECK(tgamma(x))
125 return tgamma(ex_to_numeric(x));
129 /** Evaluation of tgamma(x), the true Gamma function. Knows about integer
130 * arguments, half-integer arguments and that's it. Somebody ought to provide
131 * some good numerical evaluation some day...
133 * @exception pole_error("tgamma_eval(): simple pole",0) */
134 static ex tgamma_eval(const ex & x)
136 if (x.info(info_flags::numeric)) {
137 // trap integer arguments:
138 if (x.info(info_flags::integer)) {
139 // tgamma(n) -> (n-1)! for postitive n
140 if (x.info(info_flags::posint)) {
141 return factorial(ex_to_numeric(x).sub(_num1()));
143 throw (pole_error("tgamma_eval(): simple pole",1));
146 // trap half integer arguments:
147 if ((x*2).info(info_flags::integer)) {
148 // trap positive x==(n+1/2)
149 // tgamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n)
150 if ((x*_ex2()).info(info_flags::posint)) {
151 numeric n = ex_to_numeric(x).sub(_num1_2());
152 numeric coefficient = doublefactorial(n.mul(_num2()).sub(_num1()));
153 coefficient = coefficient.div(pow(_num2(),n));
154 return coefficient * pow(Pi,_ex1_2());
156 // trap negative x==(-n+1/2)
157 // tgamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1))
158 numeric n = abs(ex_to_numeric(x).sub(_num1_2()));
159 numeric coefficient = pow(_num_2(), n);
160 coefficient = coefficient.div(doublefactorial(n.mul(_num2()).sub(_num1())));;
161 return coefficient*power(Pi,_ex1_2());
164 // tgamma_evalf should be called here once it becomes available
167 return tgamma(x).hold();
171 static ex tgamma_deriv(const ex & x, unsigned deriv_param)
173 GINAC_ASSERT(deriv_param==0);
175 // d/dx tgamma(x) -> psi(x)*tgamma(x)
176 return psi(x)*tgamma(x);
180 static ex tgamma_series(const ex & arg,
181 const relational & rel,
186 // Taylor series where there is no pole falls back to psi function
188 // On a pole at -m use the recurrence relation
189 // tgamma(x) == tgamma(x+1) / x
190 // from which follows
191 // series(tgamma(x),x==-m,order) ==
192 // series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x==-m,order+1);
193 const ex arg_pt = arg.subs(rel);
194 if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
195 throw do_taylor(); // caught by function::series()
196 // if we got here we have to care for a simple pole at -m:
197 numeric m = -ex_to_numeric(arg_pt);
198 ex ser_denom = _ex1();
199 for (numeric p; p<=m; ++p)
201 return (tgamma(arg+m+_ex1())/ser_denom).series(rel, order+1, options);
205 REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval).
206 evalf_func(tgamma_evalf).
207 derivative_func(tgamma_deriv).
208 series_func(tgamma_series));
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 numeric nx(ex_to_numeric(x));
233 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));
323 // Psi-function (aka digamma-function)
326 static ex psi1_evalf(const ex & x)
330 END_TYPECHECK(psi(x))
332 return psi(ex_to_numeric(x));
335 /** Evaluation of digamma-function psi(x).
336 * Somebody ought to provide some good numerical evaluation some day... */
337 static ex psi1_eval(const ex & x)
339 if (x.info(info_flags::numeric)) {
340 numeric nx = ex_to_numeric(x);
341 if (nx.is_integer()) {
343 if (nx.is_positive()) {
344 // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - Euler
346 for (numeric i(nx+_num_1()); i.is_positive(); --i)
350 // for non-positive integers there is a pole:
351 throw (pole_error("psi_eval(): simple pole",1));
354 if ((_num2()*nx).is_integer()) {
356 if (nx.is_positive()) {
357 // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - Euler - 2log(2)
359 for (numeric i((nx+_num_1())*_num2()); i.is_positive(); i-=_num2())
360 rat += _num2()*i.inverse();
361 return rat-Euler-_ex2()*log(_ex2());
363 // use the recurrence relation
364 // psi(-m-1/2) == psi(-m-1/2+1) - 1 / (-m-1/2)
365 // to relate psi(-m-1/2) to psi(1/2):
366 // psi(-m-1/2) == psi(1/2) + r
367 // where r == ((-1/2)^(-1) + ... + (-m-1/2)^(-1))
369 for (numeric p(nx); p<0; ++p)
370 recur -= pow(p, _num_1());
371 return recur+psi(_ex1_2());
374 // psi1_evalf should be called here once it becomes available
377 return psi(x).hold();
380 static ex psi1_deriv(const ex & x, unsigned deriv_param)
382 GINAC_ASSERT(deriv_param==0);
384 // d/dx psi(x) -> psi(1,x)
385 return psi(_ex1(), x);
388 static ex psi1_series(const ex & arg,
389 const relational & rel,
394 // Taylor series where there is no pole falls back to polygamma function
396 // On a pole at -m use the recurrence relation
397 // psi(x) == psi(x+1) - 1/z
398 // from which follows
399 // series(psi(x),x==-m,order) ==
400 // series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x==-m,order);
401 const ex arg_pt = arg.subs(rel);
402 if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
403 throw do_taylor(); // caught by function::series()
404 // if we got here we have to care for a simple pole at -m:
405 numeric m = -ex_to_numeric(arg_pt);
407 for (numeric p; p<=m; ++p)
408 recur += power(arg+p,_ex_1());
409 return (psi(arg+m+_ex1())-recur).series(rel, order, options);
412 const unsigned function_index_psi1 =
413 function::register_new(function_options("psi").
414 eval_func(psi1_eval).
415 evalf_func(psi1_evalf).
416 derivative_func(psi1_deriv).
417 series_func(psi1_series).
421 // Psi-functions (aka polygamma-functions) psi(0,x)==psi(x)
424 static ex psi2_evalf(const ex & n, const ex & x)
429 END_TYPECHECK(psi(n,x))
431 return psi(ex_to_numeric(n), ex_to_numeric(x));
434 /** Evaluation of polygamma-function psi(n,x).
435 * Somebody ought to provide some good numerical evaluation some day... */
436 static ex psi2_eval(const ex & n, const ex & x)
438 // psi(0,x) -> psi(x)
441 // psi(-1,x) -> log(tgamma(x))
442 if (n.is_equal(_ex_1()))
443 return log(tgamma(x));
444 if (n.info(info_flags::numeric) && n.info(info_flags::posint) &&
445 x.info(info_flags::numeric)) {
446 numeric nn = ex_to_numeric(n);
447 numeric nx = ex_to_numeric(x);
448 if (nx.is_integer()) {
450 if (nx.is_equal(_num1()))
451 // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1)
452 return pow(_num_1(),nn+_num1())*factorial(nn)*zeta(ex(nn+_num1()));
453 if (nx.is_positive()) {
454 // use the recurrence relation
455 // psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1)
456 // to relate psi(n,m) to psi(n,1):
457 // psi(n,m) == psi(n,1) + r
458 // where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1))
460 for (numeric p(1); p<nx; ++p)
461 recur += pow(p, -nn+_num_1());
462 recur *= factorial(nn)*pow(_num_1(), nn);
463 return recur+psi(n,_ex1());
465 // for non-positive integers there is a pole:
466 throw (pole_error("psi2_eval(): pole",1));
469 if ((_num2()*nx).is_integer()) {
471 if (nx.is_equal(_num1_2()))
472 // use psi(n,1/2) == (-)^(n+1) * n! * (2^(n+1)-1) * zeta(n+1)
473 return pow(_num_1(),nn+_num1())*factorial(nn)*(pow(_num2(),nn+_num1()) + _num_1())*zeta(ex(nn+_num1()));
474 if (nx.is_positive()) {
475 numeric m = nx - _num1_2();
476 // use the multiplication formula
477 // psi(n,2*m) == (psi(n,m) + psi(n,m+1/2)) / 2^(n+1)
478 // to revert to positive integer case
479 return psi(n,_num2()*m)*pow(_num2(),nn+_num1())-psi(n,m);
481 // use the recurrence relation
482 // psi(n,-m-1/2) == psi(n,-m-1/2+1) - (-)^n * n! / (-m-1/2)^(n+1)
483 // to relate psi(n,-m-1/2) to psi(n,1/2):
484 // psi(n,-m-1/2) == psi(n,1/2) + r
485 // where r == (-)^(n+1) * n! * ((-1/2)^(-n-1) + ... + (-m-1/2)^(-n-1))
487 for (numeric p(nx); p<0; ++p)
488 recur += pow(p, -nn+_num_1());
489 recur *= factorial(nn)*pow(_num_1(), nn+_num_1());
490 return recur+psi(n,_ex1_2());
493 // psi2_evalf should be called here once it becomes available
496 return psi(n, x).hold();
499 static ex psi2_deriv(const ex & n, const ex & x, unsigned deriv_param)
501 GINAC_ASSERT(deriv_param<2);
503 if (deriv_param==0) {
505 throw(std::logic_error("cannot diff psi(n,x) with respect to n"));
507 // d/dx psi(n,x) -> psi(n+1,x)
508 return psi(n+_ex1(), x);
511 static ex psi2_series(const ex & n,
513 const relational & rel,
518 // Taylor series where there is no pole falls back to polygamma function
520 // On a pole at -m use the recurrence relation
521 // psi(n,x) == psi(n,x+1) - (-)^n * n! / x^(n+1)
522 // from which follows
523 // series(psi(x),x==-m,order) ==
524 // series(psi(x+m+1) - (-1)^n * n! * ((x)^(-n-1) + (x+1)^(-n-1) + ...
525 // ... + (x+m)^(-n-1))),x==-m,order);
526 const ex arg_pt = arg.subs(rel);
527 if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
528 throw do_taylor(); // caught by function::series()
529 // if we got here we have to care for a pole of order n+1 at -m:
530 numeric m = -ex_to_numeric(arg_pt);
532 for (numeric p; p<=m; ++p)
533 recur += power(arg+p,-n+_ex_1());
534 recur *= factorial(n)*power(_ex_1(),n);
535 return (psi(n, arg+m+_ex1())-recur).series(rel, order, options);
538 const unsigned function_index_psi2 =
539 function::register_new(function_options("psi").
540 eval_func(psi2_eval).
541 evalf_func(psi2_evalf).
542 derivative_func(psi2_deriv).
543 series_func(psi2_series).