* Analytic continuation for atan2(y,x) for complex y and x.

diff --git a/ginac/inifcns_trans.cpp b/ginac/inifcns_trans.cpp
index f0d785904176bd4519b6d35f127228d980e93d2e..90380efe83e1b12a5098338a8e33033e8b0474f2 100644 (file)
--- a/ginac/inifcns_trans.cpp
+++ b/ginac/inifcns_trans.cpp
@@ -830,9 +830,9 @@ static ex atan2_eval(const ex & y, const ex & x)
                if (x.info(info_flags::positive))
                        return _ex0;
 
                if (x.info(info_flags::positive))
                        return _ex0;
 

-               // atan(0, x), x real and negative -> -Pi
+               // atan(0, x), x real and negative -> Pi

                if (x.info(info_flags::negative))
                if (x.info(info_flags::negative))

-                       return _ex_1*Pi;
+                       return Pi;

        }
 
        if (x.is_zero()) {
        }
 
        if (x.is_zero()) {


@@ -869,15 +869,15 @@ static ex atan2_eval(const ex & y, const ex & x)
        }
 
        // atan(float, float) -> float
        }
 
        // atan(float, float) -> float

-       if (is_a<numeric>(y) && is_a<numeric>(x) && !y.info(info_flags::crational)
-                       && !x.info(info_flags::crational))
+       if (is_a<numeric>(y) && !y.info(info_flags::crational) &&
+           is_a<numeric>(x) && !x.info(info_flags::crational))

                return atan(ex_to<numeric>(y), ex_to<numeric>(x));
 
        // atan(real, real) -> atan(y/x) +/- Pi
        if (y.info(info_flags::real) && x.info(info_flags::real)) {
                if (x.info(info_flags::positive))
                        return atan(y/x);
                return atan(ex_to<numeric>(y), ex_to<numeric>(x));
 
        // atan(real, real) -> atan(y/x) +/- Pi
        if (y.info(info_flags::real) && x.info(info_flags::real)) {
                if (x.info(info_flags::positive))
                        return atan(y/x);

-               else if(y.info(info_flags::positive))
+               else if (y.info(info_flags::positive))

                        return atan(y/x)+Pi;
                else
                        return atan(y/x)-Pi;
                        return atan(y/x)+Pi;
                else
                        return atan(y/x)-Pi;

diff --git a/ginac/numeric.cpp b/ginac/numeric.cpp
index e4b1f22942d7e4e86ec242bcd2fcd2a2b1d719ca..1858cbd9efe927533347c5fee45160f56d496758 100644 (file)
--- a/ginac/numeric.cpp
+++ b/ginac/numeric.cpp
@@ -1397,11 +1397,11 @@ const numeric acos(const numeric &x)
 }
        
 
 }
        
 

-/** Arcustangent.
+/** Numeric arcustangent.

  *
  *  @param x complex number
  *  @return atan(x)
  *
  *  @param x complex number
  *  @return atan(x)

- *  @exception pole_error("atan(): logarithmic pole",0) */
+ *  @exception pole_error("atan(): logarithmic pole",0) if x==I or x==-I. */

 const numeric atan(const numeric &x)
 {
        if (!x.is_real() &&
 const numeric atan(const numeric &x)
 {
        if (!x.is_real() &&


@@ -1412,18 +1412,36 @@ const numeric atan(const numeric &x)
 }
 
 
 }
 
 

-/** Arcustangent.
+/** Numeric arcustangent of two arguments, analytically continued in a suitable way.

  *
  *

- *  @param x real number
- *  @param y real number
- *  @return atan(y/x) */
+ *  @param y complex number
+ *  @param x complex number
+ *  @return -I*log((x+I*y)/sqrt(x^2+y^2)), which is equal to atan(y/x) if y and
+ *    x are both real.
+ *  @exception pole_error("atan(): logarithmic pole",0) if y/x==+I or y/x==-I. */

 const numeric atan(const numeric &y, const numeric &x)
 {
 const numeric atan(const numeric &y, const numeric &x)
 {

+       if (x.is_zero() && y.is_zero())
+               return *_num0_p;

        if (x.is_real() && y.is_real())
                return cln::atan(cln::the<cln::cl_R>(x.to_cl_N()),
                                 cln::the<cln::cl_R>(y.to_cl_N()));
        if (x.is_real() && y.is_real())
                return cln::atan(cln::the<cln::cl_R>(x.to_cl_N()),
                                 cln::the<cln::cl_R>(y.to_cl_N()));

-       else
-               throw std::invalid_argument("atan(): complex argument");        
+
+       // Compute -I*log((x+I*y)/sqrt(x^2+y^2))
+       //      == -I*log((x+I*y)/sqrt((x+I*y)*(x-I*y)))
+       // Do not "simplify" this to -I/2*log((x+I*y)/(x-I*y))) or likewise.
+       // The branch cuts are easily messed up.
+       const cln::cl_N aux_p = x.to_cl_N()+cln::complex(0,1)*y.to_cl_N();
+       if (cln::zerop(aux_p)) {
+               // x+I*y==0 => y/x==I, so this is a pole (we have x!=0).
+               throw pole_error("atan(): logarithmic pole",0);
+       }
+       const cln::cl_N aux_m = x.to_cl_N()-cln::complex(0,1)*y.to_cl_N();
+       if (cln::zerop(aux_m)) {
+               // x-I*y==0 => y/x==-I, so this is a pole (we have x!=0).
+               throw pole_error("atan(): logarithmic pole",0);
+       }
+       return cln::complex(0,-1)*cln::log(aux_p/cln::sqrt(aux_p*aux_m));

 }
 
 
 }