X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_gamma.cpp;h=dfc5ee175b41f015198b2091daf1f9eff35321df;hp=9e96fda8ae83646c7e2f6b2799b819bd79f15246;hb=5f1c3b3861f1fc7978b5f734e6e058ba95de355c;hpb=97af29c12bb3074cfb4e674d71000f0712c51ba2 diff --git a/ginac/inifcns_gamma.cpp b/ginac/inifcns_gamma.cpp index 9e96fda8..dfc5ee17 100644 --- a/ginac/inifcns_gamma.cpp +++ b/ginac/inifcns_gamma.cpp @@ -4,7 +4,7 @@ * some related stuff. */ /* - * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by @@ -25,211 +25,303 @@ #include #include "inifcns.h" -#include "ex.h" #include "constant.h" #include "pseries.h" #include "numeric.h" #include "power.h" #include "relational.h" #include "symbol.h" +#include "symmetry.h" #include "utils.h" -#ifndef NO_NAMESPACE_GINAC namespace GiNaC { -#endif // ndef NO_NAMESPACE_GINAC ////////// -// Gamma-function +// Logarithm of Gamma function ////////// -static ex gamma_evalf(const ex & x) +static ex lgamma_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(gamma(x)) - - return gamma(ex_to_numeric(x)); + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(lgamma(x)) + + return lgamma(ex_to_numeric(x)); } -/** Evaluation of gamma(x). Knows about integer arguments, half-integer - * arguments and that's it. Somebody ought to provide some good numerical - * evaluation some day... +/** Evaluation of lgamma(x), the natural logarithm of the Gamma function. + * Knows about integer arguments and that's it. Somebody ought to provide + * some good numerical evaluation some day... * - * @exception std::domain_error("gamma_eval(): simple pole") */ -static ex gamma_eval(const ex & x) + * @exception GiNaC::pole_error("lgamma_eval(): logarithmic pole",0) */ +static ex lgamma_eval(const ex & x) { - if (x.info(info_flags::numeric)) { - // trap integer arguments: - if (x.info(info_flags::integer)) { - // gamma(n+1) -> n! for postitive n - if (x.info(info_flags::posint)) { - return factorial(ex_to_numeric(x).sub(_num1())); - } else { - throw (std::domain_error("gamma_eval(): simple pole")); - } - } - // trap half integer arguments: - if ((x*2).info(info_flags::integer)) { - // trap positive x==(n+1/2) - // gamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n) - if ((x*_ex2()).info(info_flags::posint)) { - numeric n = ex_to_numeric(x).sub(_num1_2()); - numeric coefficient = doublefactorial(n.mul(_num2()).sub(_num1())); - coefficient = coefficient.div(pow(_num2(),n)); - return coefficient * pow(Pi,_ex1_2()); - } else { - // trap negative x==(-n+1/2) - // gamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1)) - numeric n = abs(ex_to_numeric(x).sub(_num1_2())); - numeric coefficient = pow(_num_2(), n); - coefficient = coefficient.div(doublefactorial(n.mul(_num2()).sub(_num1())));; - return coefficient*power(Pi,_ex1_2()); - } - } - // gamma_evalf should be called here once it becomes available - } - - return gamma(x).hold(); -} + if (x.info(info_flags::numeric)) { + // trap integer arguments: + if (x.info(info_flags::integer)) { + // lgamma(n) -> log((n-1)!) for postitive n + if (x.info(info_flags::posint)) + return log(factorial(x + _ex_1())); + else + throw (pole_error("lgamma_eval(): logarithmic pole",0)); + } + // lgamma_evalf should be called here once it becomes available + } + + return lgamma(x).hold(); +} -static ex gamma_deriv(const ex & x, unsigned deriv_param) +static ex lgamma_deriv(const ex & x, unsigned deriv_param) { - GINAC_ASSERT(deriv_param==0); - - // d/dx log(gamma(x)) -> psi(x) - // d/dx gamma(x) -> psi(x)*gamma(x) - return psi(x)*gamma(x); + GINAC_ASSERT(deriv_param==0); + + // d/dx lgamma(x) -> psi(x) + return psi(x); } -static ex gamma_series(const ex & x, const symbol & s, const ex & pt, int order) +static ex lgamma_series(const ex & arg, + const relational & rel, + int order, + unsigned options) { - // method: - // Taylor series where there is no pole falls back to psi function - // evaluation. - // On a pole at -m use the recurrence relation - // gamma(x) == gamma(x+1) / x - // from which follows - // series(gamma(x),x,-m,order) == - // series(gamma(x+m+1)/(x*(x+1)*...*(x+m)),x,-m,order+1); - const ex x_pt = x.subs(s==pt); - if (!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive)) - throw do_taylor(); // caught by function::series() - // if we got here we have to care for a simple pole at -m: - numeric m = -ex_to_numeric(x_pt); - ex ser_denom = _ex1(); - for (numeric p; p<=m; ++p) - ser_denom *= x+p; - return (gamma(x+m+_ex1())/ser_denom).series(s, pt, order+1); + // method: + // Taylor series where there is no pole falls back to psi function + // evaluation. + // On a pole at -m we could use the recurrence relation + // lgamma(x) == lgamma(x+1)-log(x) + // from which follows + // series(lgamma(x),x==-m,order) == + // series(lgamma(x+m+1)-log(x)...-log(x+m)),x==-m,order); + const ex arg_pt = arg.subs(rel); + if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive)) + throw do_taylor(); // caught by function::series() + // if we got here we have to care for a simple pole of tgamma(-m): + numeric m = -ex_to_numeric(arg_pt); + ex recur; + for (numeric p; p<=m; ++p) + recur += log(arg+p); + return (lgamma(arg+m+_ex1())-recur).series(rel, order, options); } -REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_deriv, gamma_series); +REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval). + evalf_func(lgamma_evalf). + derivative_func(lgamma_deriv). + series_func(lgamma_series). + latex_name("\\log \\Gamma")); ////////// -// Beta-function +// true Gamma function +////////// + +static ex tgamma_evalf(const ex & x) +{ + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(tgamma(x)) + + return tgamma(ex_to_numeric(x)); +} + + +/** Evaluation of tgamma(x), the true Gamma function. Knows about integer + * arguments, half-integer arguments and that's it. Somebody ought to provide + * some good numerical evaluation some day... + * + * @exception pole_error("tgamma_eval(): simple pole",0) */ +static ex tgamma_eval(const ex & x) +{ + if (x.info(info_flags::numeric)) { + // trap integer arguments: + if (x.info(info_flags::integer)) { + // tgamma(n) -> (n-1)! for postitive n + if (x.info(info_flags::posint)) { + return factorial(ex_to_numeric(x).sub(_num1())); + } else { + throw (pole_error("tgamma_eval(): simple pole",1)); + } + } + // trap half integer arguments: + if ((x*2).info(info_flags::integer)) { + // trap positive x==(n+1/2) + // tgamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n) + if ((x*_ex2()).info(info_flags::posint)) { + numeric n = ex_to_numeric(x).sub(_num1_2()); + numeric coefficient = doublefactorial(n.mul(_num2()).sub(_num1())); + coefficient = coefficient.div(pow(_num2(),n)); + return coefficient * pow(Pi,_ex1_2()); + } else { + // trap negative x==(-n+1/2) + // tgamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1)) + numeric n = abs(ex_to_numeric(x).sub(_num1_2())); + numeric coefficient = pow(_num_2(), n); + coefficient = coefficient.div(doublefactorial(n.mul(_num2()).sub(_num1())));; + return coefficient*power(Pi,_ex1_2()); + } + } + // tgamma_evalf should be called here once it becomes available + } + + return tgamma(x).hold(); +} + + +static ex tgamma_deriv(const ex & x, unsigned deriv_param) +{ + GINAC_ASSERT(deriv_param==0); + + // d/dx tgamma(x) -> psi(x)*tgamma(x) + return psi(x)*tgamma(x); +} + + +static ex tgamma_series(const ex & arg, + const relational & rel, + int order, + unsigned options) +{ + // method: + // Taylor series where there is no pole falls back to psi function + // evaluation. + // On a pole at -m use the recurrence relation + // tgamma(x) == tgamma(x+1) / x + // from which follows + // series(tgamma(x),x==-m,order) == + // series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x==-m,order+1); + const ex arg_pt = arg.subs(rel); + if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive)) + throw do_taylor(); // caught by function::series() + // if we got here we have to care for a simple pole at -m: + numeric m = -ex_to_numeric(arg_pt); + ex ser_denom = _ex1(); + for (numeric p; p<=m; ++p) + ser_denom *= arg+p; + return (tgamma(arg+m+_ex1())/ser_denom).series(rel, order+1, options); +} + + +REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval). + evalf_func(tgamma_evalf). + derivative_func(tgamma_deriv). + series_func(tgamma_series). + latex_name("\\Gamma")); + + +////////// +// beta-function ////////// static ex beta_evalf(const ex & x, const ex & y) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - TYPECHECK(y,numeric) - END_TYPECHECK(beta(x,y)) - - return gamma(ex_to_numeric(x))*gamma(ex_to_numeric(y))/gamma(ex_to_numeric(x+y)); + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + TYPECHECK(y,numeric) + END_TYPECHECK(beta(x,y)) + + return tgamma(ex_to_numeric(x))*tgamma(ex_to_numeric(y))/tgamma(ex_to_numeric(x+y)); } static ex beta_eval(const ex & x, const ex & y) { - if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) { - // treat all problematic x and y that may not be passed into gamma, - // because they would throw there although beta(x,y) is well-defined - // using the formula beta(x,y) == (-1)^y * beta(1-x-y, y) - numeric nx(ex_to_numeric(x)); - numeric ny(ex_to_numeric(y)); - if (nx.is_real() && nx.is_integer() && - ny.is_real() && ny.is_integer()) { - if (nx.is_negative()) { - if (nx<=-ny) - return pow(_num_1(), ny)*beta(1-x-y, y); - else - throw (std::domain_error("beta_eval(): simple pole")); - } - if (ny.is_negative()) { - if (ny<=-nx) - return pow(_num_1(), nx)*beta(1-y-x, x); - else - throw (std::domain_error("beta_eval(): simple pole")); - } - return gamma(x)*gamma(y)/gamma(x+y); - } - // no problem in numerator, but denominator has pole: - if ((nx+ny).is_real() && - (nx+ny).is_integer() && - !(nx+ny).is_positive()) - return _ex0(); - // everything is ok: - return gamma(x)*gamma(y)/gamma(x+y); - } - - return beta(x,y).hold(); + if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) { + // treat all problematic x and y that may not be passed into tgamma, + // because they would throw there although beta(x,y) is well-defined + // using the formula beta(x,y) == (-1)^y * beta(1-x-y, y) + numeric nx(ex_to_numeric(x)); + numeric ny(ex_to_numeric(y)); + if (nx.is_real() && nx.is_integer() && + ny.is_real() && ny.is_integer()) { + if (nx.is_negative()) { + if (nx<=-ny) + return pow(_num_1(), ny)*beta(1-x-y, y); + else + throw (pole_error("beta_eval(): simple pole",1)); + } + if (ny.is_negative()) { + if (ny<=-nx) + return pow(_num_1(), nx)*beta(1-y-x, x); + else + throw (pole_error("beta_eval(): simple pole",1)); + } + return tgamma(x)*tgamma(y)/tgamma(x+y); + } + // no problem in numerator, but denominator has pole: + if ((nx+ny).is_real() && + (nx+ny).is_integer() && + !(nx+ny).is_positive()) + return _ex0(); + // everything is ok: + return tgamma(x)*tgamma(y)/tgamma(x+y); + } + + return beta(x,y).hold(); } static ex beta_deriv(const ex & x, const ex & y, unsigned deriv_param) { - GINAC_ASSERT(deriv_param<2); - ex retval; - - // d/dx beta(x,y) -> (psi(x)-psi(x+y)) * beta(x,y) - if (deriv_param==0) - retval = (psi(x)-psi(x+y))*beta(x,y); - // d/dy beta(x,y) -> (psi(y)-psi(x+y)) * beta(x,y) - if (deriv_param==1) - retval = (psi(y)-psi(x+y))*beta(x,y); - return retval; + GINAC_ASSERT(deriv_param<2); + ex retval; + + // d/dx beta(x,y) -> (psi(x)-psi(x+y)) * beta(x,y) + if (deriv_param==0) + retval = (psi(x)-psi(x+y))*beta(x,y); + // d/dy beta(x,y) -> (psi(y)-psi(x+y)) * beta(x,y) + if (deriv_param==1) + retval = (psi(y)-psi(x+y))*beta(x,y); + return retval; } -static ex beta_series(const ex & x, const ex & y, const symbol & s, const ex & pt, int order) +static ex beta_series(const ex & arg1, + const ex & arg2, + const relational & rel, + int order, + unsigned options) { - // method: - // Taylor series where there is no pole of one of the gamma functions - // falls back to beta function evaluation. Otherwise, fall back to - // gamma series directly. - // FIXME: this could need some testing, maybe it's wrong in some cases? - const ex x_pt = x.subs(s==pt); - const ex y_pt = y.subs(s==pt); - ex x_ser, y_ser, xy_ser; - if ((!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive)) && - (!y_pt.info(info_flags::integer) || y_pt.info(info_flags::positive))) - throw do_taylor(); // caught by function::series() - // trap the case where x is on a pole directly: - if (x.info(info_flags::integer) && !x.info(info_flags::positive)) - x_ser = gamma(x+s).series(s,pt,order); - else - x_ser = gamma(x).series(s,pt,order); - // trap the case where y is on a pole directly: - if (y.info(info_flags::integer) && !y.info(info_flags::positive)) - y_ser = gamma(y+s).series(s,pt,order); - else - y_ser = gamma(y).series(s,pt,order); - // trap the case where y is on a pole directly: - if ((x+y).info(info_flags::integer) && !(x+y).info(info_flags::positive)) - xy_ser = gamma(y+x+s).series(s,pt,order); - else - xy_ser = gamma(y+x).series(s,pt,order); - // compose the result: - return (x_ser*y_ser/xy_ser).series(s,pt,order); + // method: + // Taylor series where there is no pole of one of the tgamma functions + // falls back to beta function evaluation. Otherwise, fall back to + // tgamma series directly. + const ex arg1_pt = arg1.subs(rel); + const ex arg2_pt = arg2.subs(rel); + GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol)); + const symbol *s = static_cast(rel.lhs().bp); + ex arg1_ser, arg2_ser, arg1arg2_ser; + if ((!arg1_pt.info(info_flags::integer) || arg1_pt.info(info_flags::positive)) && + (!arg2_pt.info(info_flags::integer) || arg2_pt.info(info_flags::positive))) + throw do_taylor(); // caught by function::series() + // trap the case where arg1 is on a pole: + if (arg1.info(info_flags::integer) && !arg1.info(info_flags::positive)) + arg1_ser = tgamma(arg1+*s).series(rel, order, options); + else + arg1_ser = tgamma(arg1).series(rel,order); + // trap the case where arg2 is on a pole: + if (arg2.info(info_flags::integer) && !arg2.info(info_flags::positive)) + arg2_ser = tgamma(arg2+*s).series(rel, order, options); + else + arg2_ser = tgamma(arg2).series(rel,order); + // trap the case where arg1+arg2 is on a pole: + if ((arg1+arg2).info(info_flags::integer) && !(arg1+arg2).info(info_flags::positive)) + arg1arg2_ser = tgamma(arg2+arg1+*s).series(rel, order, options); + else + arg1arg2_ser = tgamma(arg2+arg1).series(rel,order); + // compose the result (expanding all the terms): + return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand(); } -REGISTER_FUNCTION(beta, beta_eval, beta_evalf, beta_deriv, beta_series); +REGISTER_FUNCTION(beta, eval_func(beta_eval). + evalf_func(beta_evalf). + derivative_func(beta_deriv). + series_func(beta_series). + latex_name("\\mbox{B}"). + set_symmetry(sy_symm(0, 1))); ////////// @@ -238,88 +330,98 @@ REGISTER_FUNCTION(beta, beta_eval, beta_evalf, beta_deriv, beta_series); static ex psi1_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(psi(x)) - - return psi(ex_to_numeric(x)); + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(psi(x)) + + return psi(ex_to_numeric(x)); } /** Evaluation of digamma-function psi(x). * Somebody ought to provide some good numerical evaluation some day... */ static ex psi1_eval(const ex & x) { - if (x.info(info_flags::numeric)) { - numeric nx = ex_to_numeric(x); - if (nx.is_integer()) { - // integer case - if (nx.is_positive()) { - // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - EulerGamma - numeric rat(0); - for (numeric i(nx+_num_1()); i.is_positive(); --i) - rat += i.inverse(); - return rat-EulerGamma; - } else { - // for non-positive integers there is a pole: - throw (std::domain_error("psi_eval(): simple pole")); - } - } - if ((_num2()*nx).is_integer()) { - // half integer case - if (nx.is_positive()) { - // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - EulerGamma - 2log(2) - numeric rat(0); - for (numeric i((nx+_num_1())*_num2()); i.is_positive(); i-=_num2()) - rat += _num2()*i.inverse(); - return rat-EulerGamma-_ex2()*log(_ex2()); - } else { - // use the recurrence relation - // psi(-m-1/2) == psi(-m-1/2+1) - 1 / (-m-1/2) - // to relate psi(-m-1/2) to psi(1/2): - // psi(-m-1/2) == psi(1/2) + r - // where r == ((-1/2)^(-1) + ... + (-m-1/2)^(-1)) - numeric recur(0); - for (numeric p(nx); p<0; ++p) - recur -= pow(p, _num_1()); - return recur+psi(_ex1_2()); - } - } - // psi1_evalf should be called here once it becomes available - } - - return psi(x).hold(); + if (x.info(info_flags::numeric)) { + numeric nx = ex_to_numeric(x); + if (nx.is_integer()) { + // integer case + if (nx.is_positive()) { + // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - Euler + numeric rat(0); + for (numeric i(nx+_num_1()); i.is_positive(); --i) + rat += i.inverse(); + return rat-Euler; + } else { + // for non-positive integers there is a pole: + throw (pole_error("psi_eval(): simple pole",1)); + } + } + if ((_num2()*nx).is_integer()) { + // half integer case + if (nx.is_positive()) { + // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - Euler - 2log(2) + numeric rat(0); + for (numeric i((nx+_num_1())*_num2()); i.is_positive(); i-=_num2()) + rat += _num2()*i.inverse(); + return rat-Euler-_ex2()*log(_ex2()); + } else { + // use the recurrence relation + // psi(-m-1/2) == psi(-m-1/2+1) - 1 / (-m-1/2) + // to relate psi(-m-1/2) to psi(1/2): + // psi(-m-1/2) == psi(1/2) + r + // where r == ((-1/2)^(-1) + ... + (-m-1/2)^(-1)) + numeric recur(0); + for (numeric p(nx); p<0; ++p) + recur -= pow(p, _num_1()); + return recur+psi(_ex1_2()); + } + } + // psi1_evalf should be called here once it becomes available + } + + return psi(x).hold(); } static ex psi1_deriv(const ex & x, unsigned deriv_param) { - GINAC_ASSERT(deriv_param==0); - - // d/dx psi(x) -> psi(1,x) - return psi(_ex1(), x); + GINAC_ASSERT(deriv_param==0); + + // d/dx psi(x) -> psi(1,x) + return psi(_ex1(), x); } -static ex psi1_series(const ex & x, const symbol & s, const ex & pt, int order) +static ex psi1_series(const ex & arg, + const relational & rel, + int order, + unsigned options) { - // method: - // Taylor series where there is no pole falls back to polygamma function - // evaluation. - // On a pole at -m use the recurrence relation - // psi(x) == psi(x+1) - 1/z - // from which follows - // series(psi(x),x,-m,order) == - // series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x,-m,order); - const ex x_pt = x.subs(s==pt); - if (!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive)) - throw do_taylor(); // caught by function::series() - // if we got here we have to care for a simple pole at -m: - numeric m = -ex_to_numeric(x_pt); - ex recur; - for (numeric p; p<=m; ++p) - recur += power(x+p,_ex_1()); - return (psi(x+m+_ex1())-recur).series(s, pt, order); + // method: + // Taylor series where there is no pole falls back to polygamma function + // evaluation. + // On a pole at -m use the recurrence relation + // psi(x) == psi(x+1) - 1/z + // from which follows + // series(psi(x),x==-m,order) == + // series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x==-m,order); + const ex arg_pt = arg.subs(rel); + if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive)) + throw do_taylor(); // caught by function::series() + // if we got here we have to care for a simple pole at -m: + numeric m = -ex_to_numeric(arg_pt); + ex recur; + for (numeric p; p<=m; ++p) + recur += power(arg+p,_ex_1()); + return (psi(arg+m+_ex1())-recur).series(rel, order, options); } -const unsigned function_index_psi1 = function::register_new("psi", psi1_eval, psi1_evalf, psi1_deriv, psi1_series); +const unsigned function_index_psi1 = + function::register_new(function_options("psi"). + eval_func(psi1_eval). + evalf_func(psi1_evalf). + derivative_func(psi1_deriv). + series_func(psi1_series). + latex_name("\\psi"). + overloaded(2)); ////////// // Psi-functions (aka polygamma-functions) psi(0,x)==psi(x) @@ -327,116 +429,126 @@ const unsigned function_index_psi1 = function::register_new("psi", psi1_eval, ps static ex psi2_evalf(const ex & n, const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(n,numeric) - TYPECHECK(x,numeric) - END_TYPECHECK(psi(n,x)) - - return psi(ex_to_numeric(n), ex_to_numeric(x)); + BEGIN_TYPECHECK + TYPECHECK(n,numeric) + TYPECHECK(x,numeric) + END_TYPECHECK(psi(n,x)) + + return psi(ex_to_numeric(n), ex_to_numeric(x)); } /** Evaluation of polygamma-function psi(n,x). * Somebody ought to provide some good numerical evaluation some day... */ static ex psi2_eval(const ex & n, const ex & x) { - // psi(0,x) -> psi(x) - if (n.is_zero()) - return psi(x); - // psi(-1,x) -> log(gamma(x)) - if (n.is_equal(_ex_1())) - return log(gamma(x)); - if (n.info(info_flags::numeric) && n.info(info_flags::posint) && - x.info(info_flags::numeric)) { - numeric nn = ex_to_numeric(n); - numeric nx = ex_to_numeric(x); - if (nx.is_integer()) { - // integer case - if (nx.is_equal(_num1())) - // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1) - return pow(_num_1(),nn+_num1())*factorial(nn)*zeta(ex(nn+_num1())); - if (nx.is_positive()) { - // use the recurrence relation - // psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1) - // to relate psi(n,m) to psi(n,1): - // psi(n,m) == psi(n,1) + r - // where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1)) - numeric recur(0); - for (numeric p(1); p psi(x) + if (n.is_zero()) + return psi(x); + // psi(-1,x) -> log(tgamma(x)) + if (n.is_equal(_ex_1())) + return log(tgamma(x)); + if (n.info(info_flags::numeric) && n.info(info_flags::posint) && + x.info(info_flags::numeric)) { + numeric nn = ex_to_numeric(n); + numeric nx = ex_to_numeric(x); + if (nx.is_integer()) { + // integer case + if (nx.is_equal(_num1())) + // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1) + return pow(_num_1(),nn+_num1())*factorial(nn)*zeta(ex(nn+_num1())); + if (nx.is_positive()) { + // use the recurrence relation + // psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1) + // to relate psi(n,m) to psi(n,1): + // psi(n,m) == psi(n,1) + r + // where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1)) + numeric recur(0); + for (numeric p(1); p psi(n+1,x) - return psi(n+_ex1(), x); + GINAC_ASSERT(deriv_param<2); + + if (deriv_param==0) { + // d/dn psi(n,x) + throw(std::logic_error("cannot diff psi(n,x) with respect to n")); + } + // d/dx psi(n,x) -> psi(n+1,x) + return psi(n+_ex1(), x); } -static ex psi2_series(const ex & n, const ex & x, const symbol & s, const ex & pt, int order) +static ex psi2_series(const ex & n, + const ex & arg, + const relational & rel, + int order, + unsigned options) { - // method: - // Taylor series where there is no pole falls back to polygamma function - // evaluation. - // On a pole at -m use the recurrence relation - // psi(n,x) == psi(n,x+1) - (-)^n * n! / x^(n+1) - // from which follows - // series(psi(x),x,-m,order) == - // series(psi(x+m+1) - (-1)^n * n! * ((x)^(-n-1) + (x+1)^(-n-1) + ... - // ... + (x+m)^(-n-1))),x,-m,order); - const ex x_pt = x.subs(s==pt); - if (!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive)) - throw do_taylor(); // caught by function::series() - // if we got here we have to care for a pole of order n+1 at -m: - numeric m = -ex_to_numeric(x_pt); - ex recur; - for (numeric p; p<=m; ++p) - recur += power(x+p,-n+_ex_1()); - recur *= factorial(n)*power(_ex_1(),n); - return (psi(n, x+m+_ex1())-recur).series(s, pt, order); + // method: + // Taylor series where there is no pole falls back to polygamma function + // evaluation. + // On a pole at -m use the recurrence relation + // psi(n,x) == psi(n,x+1) - (-)^n * n! / x^(n+1) + // from which follows + // series(psi(x),x==-m,order) == + // series(psi(x+m+1) - (-1)^n * n! * ((x)^(-n-1) + (x+1)^(-n-1) + ... + // ... + (x+m)^(-n-1))),x==-m,order); + const ex arg_pt = arg.subs(rel); + if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive)) + throw do_taylor(); // caught by function::series() + // if we got here we have to care for a pole of order n+1 at -m: + numeric m = -ex_to_numeric(arg_pt); + ex recur; + for (numeric p; p<=m; ++p) + recur += power(arg+p,-n+_ex_1()); + recur *= factorial(n)*power(_ex_1(),n); + return (psi(n, arg+m+_ex1())-recur).series(rel, order, options); } -const unsigned function_index_psi2 = function::register_new("psi", psi2_eval, psi2_evalf, psi2_deriv, psi2_series); +const unsigned function_index_psi2 = + function::register_new(function_options("psi"). + eval_func(psi2_eval). + evalf_func(psi2_evalf). + derivative_func(psi2_deriv). + series_func(psi2_series). + latex_name("\\psi"). + overloaded(2)); + -#ifndef NO_NAMESPACE_GINAC } // namespace GiNaC -#endif // ndef NO_NAMESPACE_GINAC