X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_gamma.cpp;h=c9adc19967ad37a4da71f91b1929103a0cd58586;hp=8fa1998b868541e7a77a832cec92f0182ad9611a;hb=d67dadd063fbae8e9a64560d2ea97c7af0248203;hpb=a4294a62590d45ad81e66ce59101982dc83dba51

diff --git a/ginac/inifcns_gamma.cpp b/ginac/inifcns_gamma.cpp
index 8fa1998b..c9adc199 100644
--- a/ginac/inifcns_gamma.cpp
+++ b/ginac/inifcns_gamma.cpp
@@ -4,7 +4,7 @@
  *  some related stuff. */
 
 /*
- *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2005 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
@@ -18,7 +18,7 @@
  *
  *  You should have received a copy of the GNU General Public License
  *  along with this program; if not, write to the Free Software
- *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
+ *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  */
 
 #include <vector>
@@ -30,7 +30,9 @@
 #include "numeric.h"
 #include "power.h"
 #include "relational.h"
+#include "operators.h"
 #include "symbol.h"
+#include "symmetry.h"
 #include "utils.h"
 
 namespace GiNaC {
@@ -41,11 +43,13 @@ namespace GiNaC {
 
 static ex lgamma_evalf(const ex & x)
 {
-	BEGIN_TYPECHECK
-		TYPECHECK(x,numeric)
-	END_TYPECHECK(lgamma(x))
+	if (is_exactly_a<numeric>(x)) {
+		try {
+			return lgamma(ex_to<numeric>(x));
+		} catch (const dunno &e) { }
+	}
 	
-	return lgamma(ex_to_numeric(x));
+	return lgamma(x).hold();
 }
 
 
@@ -61,7 +65,7 @@ static ex lgamma_eval(const ex & x)
 		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()));
+				return log(factorial(x + _ex_1));
 			else
 				throw (pole_error("lgamma_eval(): logarithmic pole",0));
 		}
@@ -94,15 +98,15 @@ static ex lgamma_series(const ex & arg,
 	// 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);
+	const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 	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);
+	numeric m = -ex_to<numeric>(arg_pt);
 	ex recur;
-	for (numeric p; p<=m; ++p)
+	for (numeric p = 0; p<=m; ++p)
 		recur += log(arg+p);
-	return (lgamma(arg+m+_ex1())-recur).series(rel, order, options);
+	return (lgamma(arg+m+_ex1)-recur).series(rel, order, options);
 }
 
 
@@ -119,11 +123,13 @@ REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval).
 
 static ex tgamma_evalf(const ex & x)
 {
-	BEGIN_TYPECHECK
-		TYPECHECK(x,numeric)
-	END_TYPECHECK(tgamma(x))
+	if (is_exactly_a<numeric>(x)) {
+		try {
+			return tgamma(ex_to<numeric>(x));
+		} catch (const dunno &e) { }
+	}
 	
-	return tgamma(ex_to_numeric(x));
+	return tgamma(x).hold();
 }
 
 
@@ -136,30 +142,27 @@ static ex tgamma_eval(const ex & x)
 {
 	if (x.info(info_flags::numeric)) {
 		// trap integer arguments:
-		if (x.info(info_flags::integer)) {
+		const numeric two_x = (*_num2_p)*ex_to<numeric>(x);
+		if (two_x.is_even()) {
 			// tgamma(n) -> (n-1)! for postitive n
-			if (x.info(info_flags::posint)) {
-				return factorial(ex_to_numeric(x).sub(_num1()));
+			if (two_x.is_positive()) {
+				return factorial(ex_to<numeric>(x).sub(*_num1_p));
 			} else {
 				throw (pole_error("tgamma_eval(): simple pole",1));
 			}
 		}
 		// trap half integer arguments:
-		if ((x*2).info(info_flags::integer)) {
+		if (two_x.is_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());
+			if (two_x.is_positive()) {
+				const numeric n = ex_to<numeric>(x).sub(*_num1_2_p);
+				return (doublefactorial(n.mul(*_num2_p).sub(*_num1_p)).div(pow(*_num2_p,n))) * sqrt(Pi);
 			} 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());
+				const numeric n = abs(ex_to<numeric>(x).sub(*_num1_2_p));
+				return (pow(*_num_2_p, n).div(doublefactorial(n.mul(*_num2_p).sub(*_num1_p))))*sqrt(Pi);
 			}
 		}
 		//  tgamma_evalf should be called here once it becomes available
@@ -190,16 +193,16 @@ static ex tgamma_series(const ex & arg,
 	//   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);
+	//   series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x==-m,order);
+	const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 	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();
+	const 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);
+	return (tgamma(arg+m+_ex1)/ser_denom).series(rel, order, options);
 }
 
 
@@ -216,34 +219,39 @@ REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval).
 
 static ex beta_evalf(const ex & x, const ex & y)
 {
-	BEGIN_TYPECHECK
-		TYPECHECK(x,numeric)
-		TYPECHECK(y,numeric)
-	END_TYPECHECK(beta(x,y))
+	if (is_exactly_a<numeric>(x) && is_exactly_a<numeric>(y)) {
+		try {
+			return exp(lgamma(ex_to<numeric>(x))+lgamma(ex_to<numeric>(y))-lgamma(ex_to<numeric>(x+y)));
+		} catch (const dunno &e) { }
+	}
 	
-	return tgamma(ex_to_numeric(x))*tgamma(ex_to_numeric(y))/tgamma(ex_to_numeric(x+y));
+	return beta(x,y).hold();
 }
 
 
 static ex beta_eval(const ex & x, const ex & y)
 {
+	if (x.is_equal(_ex1))
+		return 1/y;
+	if (y.is_equal(_ex1))
+		return 1/x;
 	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));
+		const numeric &nx = ex_to<numeric>(x);
+		const 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);
+					return pow(*_num_1_p, 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);
+					return pow(*_num_1_p, nx)*beta(1-y-x, x);
 				else
 					throw (pole_error("beta_eval(): simple pole",1));
 			}
@@ -251,11 +259,10 @@ static ex beta_eval(const ex & x, const ex & 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);
+		    (nx+ny).is_integer() &&
+		   !(nx+ny).is_positive())
+			 return _ex0;
+		// beta_evalf should be called here once it becomes available
 	}
 	
 	return beta(x,y).hold();
@@ -287,29 +294,29 @@ static ex beta_series(const ex & arg1,
 	// 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<symbol *>(rel.lhs().bp);
+	const ex arg1_pt = arg1.subs(rel, subs_options::no_pattern);
+	const ex arg2_pt = arg2.subs(rel, subs_options::no_pattern);
+	GINAC_ASSERT(is_a<symbol>(rel.lhs()));
+	const symbol &s = ex_to<symbol>(rel.lhs());
 	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);
+		arg1_ser = tgamma(arg1+s);
 	else
-		arg1_ser = tgamma(arg1).series(rel,order);
+		arg1_ser = tgamma(arg1);
 	// 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);
+		arg2_ser = tgamma(arg2+s);
 	else
-		arg2_ser = tgamma(arg2).series(rel,order);
+		arg2_ser = tgamma(arg2);
 	// 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);
+		arg1arg2_ser = tgamma(arg2+arg1+s);
 	else
-		arg1arg2_ser = tgamma(arg2+arg1).series(rel,order);
+		arg1arg2_ser = tgamma(arg2+arg1);
 	// compose the result (expanding all the terms):
 	return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand();
 }
@@ -319,7 +326,8 @@ REGISTER_FUNCTION(beta, eval_func(beta_eval).
                         evalf_func(beta_evalf).
                         derivative_func(beta_deriv).
                         series_func(beta_series).
-                        latex_name("\\mbox{B}"));
+                        latex_name("\\mbox{B}").
+						set_symmetry(sy_symm(0, 1)));
 
 
 //////////
@@ -328,11 +336,13 @@ REGISTER_FUNCTION(beta, eval_func(beta_eval).
 
 static ex psi1_evalf(const ex & x)
 {
-	BEGIN_TYPECHECK
-		TYPECHECK(x,numeric)
-	END_TYPECHECK(psi(x))
+	if (is_exactly_a<numeric>(x)) {
+		try {
+			return psi(ex_to<numeric>(x));
+		} catch (const dunno &e) { }
+	}
 	
-	return psi(ex_to_numeric(x));
+	return psi(x).hold();
 }
 
 /** Evaluation of digamma-function psi(x).
@@ -340,13 +350,13 @@ static ex psi1_evalf(const ex & x)
 static ex psi1_eval(const ex & x)
 {
 	if (x.info(info_flags::numeric)) {
-		numeric nx = ex_to_numeric(x);
+		const 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)
+				numeric rat = 0;
+				for (numeric i(nx+(*_num_1_p)); i>0; --i)
 					rat += i.inverse();
 				return rat-Euler;
 			} else {
@@ -354,24 +364,24 @@ static ex psi1_eval(const ex & x)
 				throw (pole_error("psi_eval(): simple pole",1));
 			}
 		}
-		if ((_num2()*nx).is_integer()) {
+		if (((*_num2_p)*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());
+				numeric rat = 0;
+				for (numeric i = (nx+(*_num_1_p))*(*_num2_p); i>0; i-=(*_num2_p))
+					rat += (*_num2_p)*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());
+				numeric recur = 0;
+				for (numeric p = nx; p<0; ++p)
+					recur -= pow(p, *_num_1_p);
+				return recur+psi(_ex1_2);
 			}
 		}
 		//  psi1_evalf should be called here once it becomes available
@@ -385,7 +395,7 @@ 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);
+	return psi(_ex1, x);
 }
 
 static ex psi1_series(const ex & arg,
@@ -401,19 +411,19 @@ static ex psi1_series(const ex & arg,
 	// 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);
+	const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 	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);
+	const 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);
+		recur += power(arg+p,_ex_1);
+	return (psi(arg+m+_ex1)-recur).series(rel, order, options);
 }
 
-const unsigned function_index_psi1 =
-	function::register_new(function_options("psi").
+unsigned psi1_SERIAL::serial =
+	function::register_new(function_options("psi", 1).
 	                       eval_func(psi1_eval).
 	                       evalf_func(psi1_evalf).
 	                       derivative_func(psi1_deriv).
@@ -427,12 +437,13 @@ const unsigned function_index_psi1 =
 
 static ex psi2_evalf(const ex & n, const ex & x)
 {
-	BEGIN_TYPECHECK
-		TYPECHECK(n,numeric)
-		TYPECHECK(x,numeric)
-	END_TYPECHECK(psi(n,x))
+	if (is_exactly_a<numeric>(n) && is_exactly_a<numeric>(x)) {
+		try {
+			return psi(ex_to<numeric>(n),ex_to<numeric>(x));
+		} catch (const dunno &e) { }
+	}
 	
-	return psi(ex_to_numeric(n), ex_to_numeric(x));
+	return psi(n,x).hold();
 }
 
 /** Evaluation of polygamma-function psi(n,x). 
@@ -443,55 +454,55 @@ static ex psi2_eval(const ex & n, const ex & x)
 	if (n.is_zero())
 		return psi(x);
 	// psi(-1,x) -> log(tgamma(x))
-	if (n.is_equal(_ex_1()))
+	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);
+		const numeric &nn = ex_to<numeric>(n);
+		const numeric &nx = ex_to<numeric>(x);
 		if (nx.is_integer()) {
 			// integer case 
-			if (nx.is_equal(_num1()))
+			if (nx.is_equal(*_num1_p))
 				// use psi(n,1) == (-)^(n+1) * n! * zeta(n+1)
-				return pow(_num_1(),nn+_num1())*factorial(nn)*zeta(ex(nn+_num1()));
+				return pow(*_num_1_p,nn+(*_num1_p))*factorial(nn)*zeta(ex(nn+(*_num1_p)));
 			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<nx; ++p)
-					recur += pow(p, -nn+_num_1());
-				recur *= factorial(nn)*pow(_num_1(), nn);
-				return recur+psi(n,_ex1());
+				numeric recur = 0;
+				for (numeric p = 1; p<nx; ++p)
+					recur += pow(p, -nn+(*_num_1_p));
+				recur *= factorial(nn)*pow((*_num_1_p), nn);
+				return recur+psi(n,_ex1);
 			} else {
 				// for non-positive integers there is a pole:
 				throw (pole_error("psi2_eval(): pole",1));
 			}
 		}
-		if ((_num2()*nx).is_integer()) {
+		if (((*_num2_p)*nx).is_integer()) {
 			// half integer case
-			if (nx.is_equal(_num1_2()))
+			if (nx.is_equal(*_num1_2_p))
 				// use psi(n,1/2) == (-)^(n+1) * n! * (2^(n+1)-1) * zeta(n+1)
-				return pow(_num_1(),nn+_num1())*factorial(nn)*(pow(_num2(),nn+_num1()) + _num_1())*zeta(ex(nn+_num1()));
+				return pow(*_num_1_p,nn+(*_num1_p))*factorial(nn)*(pow(*_num2_p,nn+(*_num1_p)) + (*_num_1_p))*zeta(ex(nn+(*_num1_p)));
 			if (nx.is_positive()) {
-				numeric m = nx - _num1_2();
+				const numeric m = nx - (*_num1_2_p);
 				// use the multiplication formula
 				//   psi(n,2*m) == (psi(n,m) + psi(n,m+1/2)) / 2^(n+1)
 				// to revert to positive integer case
-				return psi(n,_num2()*m)*pow(_num2(),nn+_num1())-psi(n,m);
+				return psi(n,(*_num2_p)*m)*pow((*_num2_p),nn+(*_num1_p))-psi(n,m);
 			} else {
 				// use the recurrence relation
 				//   psi(n,-m-1/2) == psi(n,-m-1/2+1) - (-)^n * n! / (-m-1/2)^(n+1)
 				// to relate psi(n,-m-1/2) to psi(n,1/2):
 				//   psi(n,-m-1/2) == psi(n,1/2) + r
 				// where r == (-)^(n+1) * n! * ((-1/2)^(-n-1) + ... + (-m-1/2)^(-n-1))
-				numeric recur(0);
-				for (numeric p(nx); p<0; ++p)
-					recur += pow(p, -nn+_num_1());
-				recur *= factorial(nn)*pow(_num_1(), nn+_num_1());
-				return recur+psi(n,_ex1_2());
+				numeric recur = 0;
+				for (numeric p = nx; p<0; ++p)
+					recur += pow(p, -nn+(*_num_1_p));
+				recur *= factorial(nn)*pow(*_num_1_p, nn+(*_num_1_p));
+				return recur+psi(n,_ex1_2);
 			}
 		}
 		//  psi2_evalf should be called here once it becomes available
@@ -509,7 +520,7 @@ static ex psi2_deriv(const ex & n, const ex & x, unsigned deriv_param)
 		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);
+	return psi(n+_ex1, x);
 }
 
 static ex psi2_series(const ex & n,
@@ -527,20 +538,20 @@ static ex psi2_series(const ex & n,
 	//   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);
+	const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 	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);
+	const 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);
+		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(function_options("psi").
+unsigned psi2_SERIAL::serial =
+	function::register_new(function_options("psi", 2).
 	                       eval_func(psi2_eval).
 	                       evalf_func(psi2_evalf).
 	                       derivative_func(psi2_deriv).