X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_gamma.cpp;h=8ed385150e8175d15df02931ecd5c8fd8796c5cc;hp=dbe8840ac66245e702e26ac4cafb02b0e3a4c46c;hb=a6bb52b00bf185271774e7d56215923700a3ec40;hpb=4afb5bbe2c0b0a60928120a042997ba7d89e8f5c

diff --git a/ginac/inifcns_gamma.cpp b/ginac/inifcns_gamma.cpp
index dbe8840a..8ed38515 100644
--- a/ginac/inifcns_gamma.cpp
+++ b/ginac/inifcns_gamma.cpp
@@ -86,7 +86,9 @@ static ex gamma_eval(const ex & x)
                 return coefficient*power(Pi,_ex1_2());
             }
         }
+        //  gamma_evalf should be called here once it becomes available
     }
+    
     return gamma(x).hold();
 }    
 
@@ -99,7 +101,7 @@ static ex gamma_diff(const ex & x, unsigned diff_param)
     return psi(x)*gamma(x);
 }
 
-static ex gamma_series(const ex & x, const symbol & s, const ex & point, int order)
+static ex gamma_series(const ex & x, const symbol & s, const ex & pt, int order)
 {
     // method:
     // Taylor series where there is no pole falls back to psi function
@@ -108,17 +110,17 @@ static ex gamma_series(const ex & x, const symbol & s, const ex & point, int ord
     //   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:
-    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;
-    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);
@@ -141,11 +143,11 @@ static ex beta_evalf(const ex & x, const ex & y)
 static ex beta_eval(const ex & x, const ex & y)
 {
     if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
-        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
         // 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()) {
@@ -188,29 +190,36 @@ static ex beta_diff(const ex & x, const ex & y, unsigned diff_param)
     return retval;
 }
 
-static ex beta_series(const ex & x, const ex & y, const symbol & s, const ex & point, int order)
+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 falls back to beta 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);
-    ex xpoint = x.subs(s==point);
-    ex ypoint = y.subs(s==point);
-    if ((!xpoint.info(info_flags::integer) || xpoint.info(info_flags::positive)) &&
-        (!ypoint.info(info_flags::integer) || ypoint.info(info_flags::positive)))
+    // 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()
-    // if we got here we have to care for a simple pole at -m:
-    throw (std::domain_error("beta_series(): please code me"));
-    /*numeric m = -ex_to_numeric(xpoint);
-     *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);*/
+    // 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);
@@ -233,28 +242,43 @@ static ex psi1_evalf(const ex & x)
 static ex psi1_eval(const ex & x)
 {
     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);
-                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;
+            } 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);
-                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();
 }
 
@@ -266,7 +290,7 @@ static ex psi1_diff(const ex & x, unsigned diff_param)
     return psi(_ex1(), x);
 }
 
-static ex psi1_series(const ex & x, const symbol & s, const ex & 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
@@ -276,15 +300,15 @@ static ex psi1_series(const ex & x, const symbol & s, const ex & 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);
-    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:
-    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());
-    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);
@@ -318,24 +342,53 @@ static ex psi2_eval(const ex & n, const ex & x)
         numeric nn = ex_to_numeric(n);
         numeric nx = ex_to_numeric(x);
         if (nx.is_integer()) {
+            // integer case 
             if (nx.is_equal(_num1()))
-                return pow(_num_1(), nn+_num1())*factorial(nn)*zeta(ex(nn+_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;
+                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"));
             }
-            // 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();
 }    
 
@@ -351,7 +404,7 @@ static ex psi2_diff(const ex & n, const ex & x, unsigned diff_param)
     return psi(n+_ex1(), x);
 }
 
-static ex psi2_series(const ex & n, const ex & x, const symbol & s, const ex & 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
@@ -362,16 +415,16 @@ static ex psi2_series(const ex & n, const ex & x, const symbol & s, const ex & p
     //   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))
+    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(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);
-    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);