- speedup by declaring x_pt and so on const

[ginac.git] / ginac / inifcns_gamma.cpp

diff --git a/ginac/inifcns_gamma.cpp b/ginac/inifcns_gamma.cpp
index 3f7fc677adda5d3725418a02be8351bc72f4001c..8ed385150e8175d15df02931ecd5c8fd8796c5cc 100644 (file)
--- a/ginac/inifcns_gamma.cpp
+++ b/ginac/inifcns_gamma.cpp
@@ -42,7 +42,7 @@ namespace GiNaC {
 // Gamma-function
 //////////
 
 // Gamma-function
 //////////
 

-static ex gamma_evalf(ex const & x)
+static ex gamma_evalf(const ex & x)

 {
     BEGIN_TYPECHECK
         TYPECHECK(x,numeric)
 {
     BEGIN_TYPECHECK
         TYPECHECK(x,numeric)


@@ -56,7 +56,7 @@ static ex gamma_evalf(ex const & x)
  *  evaluation some day...
  *
  *  @exception std::domain_error("gamma_eval(): simple pole") */
  *  evaluation some day...
  *
  *  @exception std::domain_error("gamma_eval(): simple pole") */

-static ex gamma_eval(ex const & x)
+static ex gamma_eval(const ex & x)

 {
     if (x.info(info_flags::numeric)) {
         // trap integer arguments:
 {
     if (x.info(info_flags::numeric)) {
         // trap integer arguments:


@@ -72,25 +72,27 @@ static ex gamma_eval(ex const & x)
         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*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*2).info(info_flags::posint)) {
+            if ((x*_ex2()).info(info_flags::posint)) {

                 numeric n = ex_to_numeric(x).sub(_num1_2());
                 numeric coefficient = doublefactorial(n.mul(_num2()).sub(_num1()));
                 numeric n = ex_to_numeric(x).sub(_num1_2());
                 numeric coefficient = doublefactorial(n.mul(_num2()).sub(_num1()));

-                coefficient = coefficient.div(_num2().power(n));
-                return coefficient * pow(Pi,_num1_2());
+                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()));
             } 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 = numeric(-2).power(n);
+                numeric coefficient = pow(_num_2(), n);

                 coefficient = coefficient.div(doublefactorial(n.mul(_num2()).sub(_num1())));;
                 coefficient = coefficient.div(doublefactorial(n.mul(_num2()).sub(_num1())));;

-                return coefficient*sqrt(Pi);
+                return coefficient*power(Pi,_ex1_2());

             }
         }
             }
         }

+        //  gamma_evalf should be called here once it becomes available

     }
     }

+    

     return gamma(x).hold();
 }    
 
     return gamma(x).hold();
 }    
 

-static ex gamma_diff(ex const & x, unsigned diff_param)
+static ex gamma_diff(const ex & x, unsigned diff_param)

 {
     GINAC_ASSERT(diff_param==0);
     
 {
     GINAC_ASSERT(diff_param==0);
     


@@ -99,25 +101,26 @@ static ex gamma_diff(ex const & x, unsigned diff_param)
     return psi(x)*gamma(x);
 }
 
     return psi(x)*gamma(x);
 }
 

-static ex gamma_series(ex const & x, symbol const & s, ex const & point, int order)
+static ex gamma_series(const ex & x, const symbol & s, const ex & pt, int order)

 {
     // method:
 {
     // method:

-    // Taylor series where there is no pole falls back to psi function evaluation.
+    // 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) ==
     // 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);
-    ex xpoint = x.subs(s==point);
-    if (!xpoint.info(info_flags::integer) || xpoint.info(info_flags::positive))
+    //   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:
         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(xpoint);
+    numeric m = -ex_to_numeric(x_pt);

     ex ser_numer = gamma(x+m+_ex1());
     ex ser_denom = _ex1();
     for (numeric p; p<=m; ++p)
         ser_denom *= x+p;
     ex ser_numer = gamma(x+m+_ex1());
     ex ser_denom = _ex1();
     for (numeric p; p<=m; ++p)
         ser_denom *= x+p;

-    return (ser_numer/ser_denom).series(s, point, order+1);
+    return (ser_numer/ser_denom).series(s, pt, order+1);

 }
 
 REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series);
 }
 
 REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series);


@@ -126,7 +129,7 @@ REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series);
 // Beta-function
 //////////
 
 // Beta-function
 //////////
 

-static ex beta_evalf(ex const & x, ex const & y)
+static ex beta_evalf(const ex & x, const ex & y)

 {
     BEGIN_TYPECHECK
         TYPECHECK(x,numeric)
 {
     BEGIN_TYPECHECK
         TYPECHECK(x,numeric)


@@ -137,24 +140,25 @@ static ex beta_evalf(ex const & x, ex const & y)
         / gamma(ex_to_numeric(x+y));
 }
 
         / gamma(ex_to_numeric(x+y));
 }
 

-static ex beta_eval(ex const & x, ex const & y)
+static ex beta_eval(const ex & x, const ex & y)

 {
     if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
 {
     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));
         numeric nx(ex_to_numeric(x));
         numeric ny(ex_to_numeric(y));

-        // 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:

         if (nx.is_real() && nx.is_integer() &&
             ny.is_real() && ny.is_integer()) {
             if (nx.is_negative()) {
                 if (nx<=-ny)
         if (nx.is_real() && nx.is_integer() &&
             ny.is_real() && ny.is_integer()) {
             if (nx.is_negative()) {
                 if (nx<=-ny)

-                    return _num_1().power(ny)*beta(1-x-y, y);
+                    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)
                 else
                     throw (std::domain_error("beta_eval(): simple pole"));
             }
             if (ny.is_negative()) {
                 if (ny<=-nx)

-                    return _num_1().power(nx)*beta(1-y-x, x);
+                    return pow(_num_1(), nx)*beta(1-y-x, x);

                 else
                     throw (std::domain_error("beta_eval(): simple pole"));
             }
                 else
                     throw (std::domain_error("beta_eval(): simple pole"));
             }


@@ -164,13 +168,15 @@ static ex beta_eval(ex const & x, ex const & y)
         if ((nx+ny).is_real() &&
             (nx+ny).is_integer() &&
             !(nx+ny).is_positive())
         if ((nx+ny).is_real() &&
             (nx+ny).is_integer() &&
             !(nx+ny).is_positive())

-            return _ex0();
+             return _ex0();
+        // everything is ok:

         return gamma(x)*gamma(y)/gamma(x+y);
     }
         return gamma(x)*gamma(y)/gamma(x+y);
     }

+    

     return beta(x,y).hold();
 }
 
     return beta(x,y).hold();
 }
 

-static ex beta_diff(ex const & x, ex const & y, unsigned diff_param)
+static ex beta_diff(const ex & x, const ex & y, unsigned diff_param)

 {
     GINAC_ASSERT(diff_param<2);
     ex retval;
 {
     GINAC_ASSERT(diff_param<2);
     ex retval;


@@ -179,18 +185,50 @@ static ex beta_diff(ex const & x, ex const & y, unsigned diff_param)
     if (diff_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 (diff_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 (diff_param==1)  
+    if (diff_param==1)

         retval = (psi(y)-psi(x+y))*beta(x,y);
     return retval;
 }
 
         retval = (psi(y)-psi(x+y))*beta(x,y);
     return retval;
 }
 

-REGISTER_FUNCTION(beta, beta_eval, beta_evalf, beta_diff, NULL);
+static ex beta_series(const ex & x, const ex & y, const symbol & s, const ex & pt, int order)
+{
+    // 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);
+}
+
+REGISTER_FUNCTION(beta, beta_eval, beta_evalf, beta_diff, beta_series);

 
 //////////
 // Psi-function (aka digamma-function)
 //////////
 
 
 //////////
 // Psi-function (aka digamma-function)
 //////////
 

-static ex psi1_evalf(ex const & x)
+static ex psi1_evalf(const ex & x)

 {
     BEGIN_TYPECHECK
         TYPECHECK(x,numeric)
 {
     BEGIN_TYPECHECK
         TYPECHECK(x,numeric)


@@ -201,35 +239,50 @@ static ex psi1_evalf(ex const & x)
 
 /** Evaluation of digamma-function psi(x).
  *  Somebody ought to provide some good numerical evaluation some day... */
 
 /** Evaluation of digamma-function psi(x).
  *  Somebody ought to provide some good numerical evaluation some day... */

-static ex psi1_eval(ex const & x)
+static ex psi1_eval(const ex & x)

 {
     if (x.info(info_flags::numeric)) {
 {
     if (x.info(info_flags::numeric)) {

-        if (x.info(info_flags::integer) && !x.info(info_flags::positive))
-            throw (std::domain_error("psi_eval(): simple pole"));
-        if (x.info(info_flags::positive)) {
-            // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - EulerGamma
-            if (x.info(info_flags::integer)) {
+        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);
                 numeric rat(0);

-                for (numeric i(ex_to_numeric(x)-_num1()); i.is_positive(); --i)
+                for (numeric i(nx+_num_1()); i.is_positive(); --i)

                     rat += i.inverse();
                 return rat-EulerGamma;
                     rat += i.inverse();
                 return rat-EulerGamma;

+            } else {
+                // for non-positive integers there is a pole:
+                throw (std::domain_error("psi_eval(): simple pole"));

             }
             }

-            // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - EulerGamma - 2log(2)
-            if ((_ex2()*x).info(info_flags::integer)) {
+        }
+        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);
                 numeric rat(0);

-                for (numeric i((ex_to_numeric(x)-_num1())*_num2()); i.is_positive(); i-=_num2())
-                    rat += _num2()*i.inverse();
-                return rat-EulerGamma-_ex2()*log(_ex2());
-            }
-            if (x.compare(_ex1())==1) {
-                // should call numeric, since >1
+                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();
 }
 
     return psi(x).hold();
 }
 

-static ex psi1_diff(ex const & x, unsigned diff_param)
+static ex psi1_diff(const ex & x, unsigned diff_param)

 {
     GINAC_ASSERT(diff_param==0);
     
 {
     GINAC_ASSERT(diff_param==0);
     


@@ -237,7 +290,7 @@ static ex psi1_diff(ex const & x, unsigned diff_param)
     return psi(_ex1(), x);
 }
 
     return psi(_ex1(), x);
 }
 

-static ex psi1_series(ex const & x, symbol const & s, ex const & point, int order)
+static ex psi1_series(const ex & x, const symbol & s, const ex & pt, int order)

 {
     // method:
     // Taylor series where there is no pole falls back to polygamma function
 {
     // method:
     // Taylor series where there is no pole falls back to polygamma function


@@ -247,15 +300,15 @@ static ex psi1_series(ex const & x, symbol const & s, ex const & point, int orde
     // 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);
     // 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);

-    ex xpoint = x.subs(s==point);
-    if (!xpoint.info(info_flags::integer) || xpoint.info(info_flags::positive))
+    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:
         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(xpoint);
+    numeric m = -ex_to_numeric(x_pt);

     ex recur;
     for (numeric p; p<=m; ++p)
         recur += power(x+p,_ex_1());
     ex recur;
     for (numeric p; p<=m; ++p)
         recur += power(x+p,_ex_1());

-    return (psi(x+m+_ex1())-recur).series(s, point, order);
+    return (psi(x+m+_ex1())-recur).series(s, pt, order);

 }
 
 const unsigned function_index_psi1 = function::register_new("psi", psi1_eval, psi1_evalf, psi1_diff, psi1_series);
 }
 
 const unsigned function_index_psi1 = function::register_new("psi", psi1_eval, psi1_evalf, psi1_diff, psi1_series);


@@ -264,7 +317,7 @@ const unsigned function_index_psi1 = function::register_new("psi", psi1_eval, ps
 // Psi-functions (aka polygamma-functions)  psi(0,x)==psi(x)
 //////////
 
 // Psi-functions (aka polygamma-functions)  psi(0,x)==psi(x)
 //////////
 

-static ex psi2_evalf(ex const & n, ex const & x)
+static ex psi2_evalf(const ex & n, const ex & x)

 {
     BEGIN_TYPECHECK
         TYPECHECK(n,numeric)
 {
     BEGIN_TYPECHECK
         TYPECHECK(n,numeric)


@@ -276,7 +329,7 @@ static ex psi2_evalf(ex const & n, ex const & x)
 
 /** Evaluation of polygamma-function psi(n,x). 
  *  Somebody ought to provide some good numerical evaluation some day... */
 
 /** Evaluation of polygamma-function psi(n,x). 
  *  Somebody ought to provide some good numerical evaluation some day... */

-static ex psi2_eval(ex const & n, ex const & x)
+static ex psi2_eval(const ex & n, const ex & x)

 {
     // psi(0,x) -> psi(x)
     if (n.is_zero())
 {
     // psi(0,x) -> psi(x)
     if (n.is_zero())


@@ -288,13 +341,58 @@ static ex psi2_eval(ex const & n, ex const & x)
         x.info(info_flags::numeric)) {
         numeric nn = ex_to_numeric(n);
         numeric nx = ex_to_numeric(x);
         x.info(info_flags::numeric)) {
         numeric nn = ex_to_numeric(n);
         numeric nx = ex_to_numeric(x);

-        if (x.is_equal(_ex1()))
-            return _num_1().power(nn+_num1())*factorial(nn)*zeta(ex(nn+_num1()));
+        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<nx; ++p)
+                    recur += pow(p, -nn+_num_1());
+                recur *= factorial(nn)*pow(_num_1(), nn);
+                return recur+psi(n,_ex1());
+            } else {
+                // for non-positive integers there is a pole:
+                throw (std::domain_error("psi2_eval(): pole"));
+            }
+        }
+        if ((_num2()*nx).is_integer()) {
+            // half integer case
+            if (nx.is_equal(_num1_2()))
+                // 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()));
+            if (nx.is_positive()) {
+                numeric m = nx - _num1_2();
+                // 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);
+            } 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());
+            }
+        }
+        //  psi2_evalf should be called here once it becomes available

     }
     }

+    

     return psi(n, x).hold();
 }    
 
     return psi(n, x).hold();
 }    
 

-static ex psi2_diff(ex const & n, ex const & x, unsigned diff_param)
+static ex psi2_diff(const ex & n, const ex & x, unsigned diff_param)

 {
     GINAC_ASSERT(diff_param<2);
     
 {
     GINAC_ASSERT(diff_param<2);
     


@@ -303,30 +401,30 @@ static ex psi2_diff(ex const & n, ex const & x, unsigned diff_param)
         throw(std::logic_error("cannot diff psi(n,x) with respect to n"));
     }
     // d/dx psi(n,x) -> psi(n+1,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+1, x);
+    return psi(n+_ex1(), x);

 }
 
 }
 

-static ex psi2_series(ex const & n, ex const & x, symbol const & s, ex const & point, int order)
+static ex psi2_series(const ex & n, const ex & x, const symbol & s, const ex & pt, int order)

 {
     // method:
     // Taylor series where there is no pole falls back to polygamma function
     // evaluation.
     // On a pole at -m use the recurrence relation
 {
     // 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! / z^(n+1)
+    //   psi(n,x) == psi(n,x+1) - (-)^n * n! / x^(n+1)

     // from which follows
     //   series(psi(x),x,-m,order) == 
     // 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);
-    ex xpoint = x.subs(s==point);
-    if (!xpoint.info(info_flags::integer) || xpoint.info(info_flags::positive))
+    //   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:
         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(xpoint);
+    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);
     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, point, order);
+    return (psi(n, x+m+_ex1())-recur).series(s, pt, order);

 }
 
 const unsigned function_index_psi2 = function::register_new("psi", psi2_eval, psi2_evalf, psi2_diff, psi2_series);
 }
 
 const unsigned function_index_psi2 = function::register_new("psi", psi2_eval, psi2_evalf, psi2_diff, psi2_series);